Secular Dynamics in Hierarchical Three-Body Systems 3

ino rpe n ihrmlilsaogbnre ihper with binaries among fra multiples the higher that (10 found and He triples companion. of additional tion one least at have ieypoue hog rpeeouin eua ffcs( effects Secular evolution. triple through produced likely 42 mag., 10 than triple. brighter am that binaries noted contact and binaries, 151 contact of sample a surveyed have o.Nt .Ato.Soc. Astron. R. Not. Mon. mdrNaoz Smadar Systems Three-Body Hierarchical in Dynamics Secular rae.Tkvnn(97 hwdta 0 fbnr tr wi stars binary substanti of 40% actually that period is showed proportion (1997) r Tokovinin the be greater. can that we confident companions doubl or- distant sonably least and wider faint at much finding are against a stars fects on bright Eggleton of body t 1997; 50% third which (Tokovinin than a in more by Probably triples, orbited bit. hierarchical is be binary must (inner) these arguments ity rpesa ytm r eivdt evr omn(e.g., common very be to believed Eggleton are 1997; systems Tokovinin star Triple INTRODUCTION 1 2018 May 25 c † 4 3 2 1 IR,Nrhetr nvriy vntn L628 USA 60208, IL Evanston, University, Northwestern CIERA, isenFellow Einstein arXiv:1107.2414v2 [astro-ph.EP] 19 Feb 98 UPMC, CNRS, 2013 7095, UMR Paris, Univers de Northwestern d’Astrophysique Astronomy, Institut and Physics Harvard-Smithsonia of Computation, Department and Theory for Institute 00RAS 0000 − aycoeselrbnre ihtocmatojcsare objects compact two with binaries stellar close Many 0 )is d) 100 < 0di hc h rmr sadaf(0 dwarf a is primary the which in d 10 ∼ 0.Mroe,Piul n uisi(2006) Rucinski and Pribulla Moreover, 10%. 1 , 2 , † 000 ilM Farr M. Will , tal. et tal. et 0–0 00)Pitd2 a 08(NL (MN 2018 May 25 Printed (0000) 000–000 , nyi h ii nwihoeo h ne w oisi asests part test octupole massless at a evolution. studies is inclination previous bodies the two with for inner agree not the results but of evolution, our stud one circular; previous which is of in orbit those limit with the to in agree triples orientat only results stellar from its Our systems, in systems. astrophysical flips planetary of oscilla variety chaotic oute a to undergo eccentric to system an tions and a for eccentricities order, for high to octupole possibility extremely Already the the system). At including orbits. the behaviors retrograde of momen new t momentum angular show angular of the sults total treatments of the previous z-component to the some Hamiltonia parallel of in in conservation error order a hierarch an octupole assumed of corrects to dynamics derivation equations secular evolution Our the secular revisit we the Here derive o inclinations. large-amplitude exa and For time produce constant. can ities on are mechanism axes orientation, Kozai-Lidov the semimajor t and the systems, planetary shape but scales, from change time contexts, may orbital astrophysical orbits co many the triple in approximation hierarchical useful of evolution very the be for approximation secular The ABSTRACT 07.Fo yaia stabil- dynamical From 2007). 07.Gvnteslcinef- selection the Given 2007). ± 1 oa Lithwick Yoram , %aea least at are 5% . 5 − 1 . etrfrAtohsc,6 adnS. abig,M,USA MA, Cambridge, St.; Garden 60 Astrophysics, for Center n 5 M ity ally i.e., ong iod ea- ⊙ he th c- e ) i dAao -51 Paris F-75014 Arago, bd bis a etil-nue astase nabakhl-ht dw hole-white black cluster a globular in transfer in binary. mass XRBs triple-induced hole be black may for dense mechanism Blaes of formation 2002; centers Hamilton the Ivanova and cently, at blac Miller binary holes 2003; short-period black important of (Wen formation of an the growth and play clusters addition, the to star both In suggested in been 2012). role has Thompson cycling and 2009; Fabrycky impor- Shappee Kozai-Lidov Harringt and 2011; Perets an (e.g. 2007; stars Thompson as Tremaine triple and Kiseleva Fabrycky Söderhjelm proposed 1982; of 1979; 1998; Shaham been evolution and the Mazeh have 1969; in below), or element see the (Kozai tant cycling to 1962, Kozai-Lidov compared specifically Lidov long and period), timescales bital on interactions coherent seodi u w oa ytm nteasmdhierarchica assumed the in studie In an system. (1962) solar on own perturbation our Kozai in gravitational asteroid dynamics. Jupiter’s system of effects planetary the in role tant eua etrain ntil ytm lopa nimpor- an play also systems triple in perturbations Secular 1 , 3 A rdrcA Rasio A. Frederic , T E tl l v2.2) file style X tal. et 21)soe httems important most the that showed (2010) ri,teinrobtcnreach can orbit inner the orbit, r pe o ihyicie triple inclined highly for mple, egn opc iaisand binaries compact merging e oet uduoeorder quadrupole to done ies this In systems. triple-star o u fteinrobt(i.e., orbit inner the of tum epolmta implicitly that problem he clain fteeccentric- the of scillations fiuain a rvnto proven has nfigurations uduoeodr u re- our order, quadrupole 1 re o h eccentricity the for order o.W ics applica- discuss We ion. , 3 cltil ytm.We systems. triple ical etraintheory. perturbation n clslne hnthe than longer scales enTeyssandier Jean , efo rgaeto prograde from te ceadteouter the and icle tal. et 02138 02.Re- 2002). holes k clined con- l 1962; tal. et arf on d s - 1 , 4 2 Naoz et al.

figuration, treating the asteroid as a test particle, Kozai (1962) refer to this limit as the “standard” treatment of Kozai oscil- found that its inclination and eccentricity fluctuate on timescales lations, i.e. quadrupole-level approximation in the test particle much larger than its orbital period. Jupiter, assumed to be in a limit (test particle quadrupole, hereafter TPQ). circular orbit, carries most of the angular momentum of the sys- In this paper we show that a common mistake in the Hamil- tem. Due to Jupiter’s circular orbit and the negligible mass of toniano treatment of these secular systems can lead to the er- the asteroid, the system’s potential is axisymmetric and thus the roneous conclusion that the z-component of the inner orbit’s component of the inner orbit’s angular momentum along the to- angular momentum is constant outside the TPQ limit; in fact, tal angular momentum is conserved during the evolution. Kozai the z-component of the inner orbit’s angular momentum is only (1979) also showed the importance of secular interactions for conserved by the evolution in the test-particle limit and to the dynamics of comets (see also Quinn et al. 1990; Bailey et al. quadrupole order. To demonstrate the error we focus on the 1992; Thomas and Morbidelli 1996). The evolution of the orbits quadrupole (non-test-particle) approximation in the main body of binary minor planets is dominated by the secular gravita- of the paper, but we include the full octupole–order equations of tional perturbation from the sun (Perets and Naoz 2009); prop- motion in an appendix. erly accounting for the resulting secular effects—including Kozai In what follows we show the applications of these two ef- cycling—accurately reproduces the binary minor planet orbital fects (i.e., correcting the error and including the full octupole– distribution seen today (Naoz et al. 2010; Grundy et al. 2011). order equations of motion) by considering different astrophysical In addition Kinoshita and Nakai (1991), Vashkov’yak (1999), systems. Note that the applications illustrated in the text are Carruba et al. (2002), Nesvorný et al. (2003), Cuk´ and Burns inspired by real systems; however, we caution that we consider (2004) and Kinoshita and Nakai (2007) suggested that secular here only Newtonian point mass dynamics, while in reality other interactions may explain the significant inclinations of gas giant effects such as tides and general relativity can greatly effect the satellites and Jovian irregular satellites. evolution. For example, general relativity may alter the evolution Similar analyses have been applied to the orbits of extra- of the system, which can give rise to a resonant behavior of the in- solar planets (e.g., Innanen et al. 1997; Wu and Murray 2003; ner orbits eccentricity (e.g., Ford et al. 2000a; Naoz et al. 2012b). Fabrycky and Tremaine 2007; Wu et al. 2007; Naoz et al. 2011; Furthermore, tidal forces can suppress the eccentricity growth of Veras and Ford 2010; Correia et al. 2011). Naoz et al. (2011) the inner orbit, and thus significantly modify the evolutionary considered the secular evolution of a triple system consisting of track of the system (e.g., Mazeh and Shaham 1979; Söderhjelm an inner binary containing a star and a Jupiter-like planet at 1984; Kiseleva et al. 1998). In particular tides, in some cases, can several AU, orbited by a distant Jupiter-like planet or brown- considerably suppress the chaotic behavior that arises in the pres- dwarf companion. Perturbations from the outer body can drive ence of the the octupolelevel of approximation (e.g., Naoz et al. Kozai-like cycles in the inner binary, which, when planet-star 2011, 2012a). Therefore, while the examples presented in this tidal effects are incorporated, can lead to the capture of the inner paper are inspired by real astronomical systems, the true evolu- planet onto a close, highly-inclined or even retrograde orbit, sim- tionary behavior will be modified from what we show once the ilar to the orbits of the observed retrograde “hot Jupiters.” Many eccentricity becomes too high. other studies of exoplanet dynamics have considered similar sys- This paper is organized as follows. We first present the gen- tems, but with a very distant stellar binary companion acting eral framework (§2); we then derive the complete formalism for as perturber. In such systems, the outer star completely domi- the quadrupole-level approximation and the equations of motion nates the orbital angular momentum, and the problem reduces to (§3), we also develop the octupole-level approximation equations test-particle evolution (see Lithwick and Naoz 2011; Katz et al. of motion in §4. We discuss a few of the most important implica- 2011; Naoz et al. 2012a). If the lowest level of approximation is tions of the correct formalism in §5. We also compare our results applicable (e.g., the outer perturber is on a circular orbit), the with those of previous studies (§5) and offer some conclusions in z-component of the inner orbit’s angular momentum is conserved §6. (e.g., Lidov and Ziglin 1974). In early studies of high-inclination secular perturbations (Kozai 1962; Lidov 1962), the outer orbit was circular and again 2 HAMILTONIAN PERTURBATION THEORY dominated the orbital angular momentum of the system. In this FOR HIERARCHICAL TRIPLE SYSTEMS situation, the component of the inner orbit’s angular momentum along the z-axis is conserved. In many later studies the assump- Many gravitational triple systems are in a hierarchical tion that the z-component of the inner orbit’s angular momen- configuration—two objects orbit each other in a relatively tight tum is constant was built into the equations (e.g. Eggleton et al. inner binary while the third object is on a much wider orbit. If 1998; Mikkola and Tanikawa 1998; Zdziarski et al. 2007). In fact the third object is sufficiently distant, an analytic, perturbative these studies are only valid in the limit of a test particle forced approach can be used to calculate the evolution of the system. by a perturber on a circular orbit. To leading order in the ratio In the usual secular approximation (e.g., Marchal 1990), the two of semimajor axes, the double averaged potential of the outer or- orbits torque each other and exchange angular momentum, but bit is axisymmetric (even for an eccentric outer perturber), thus not energy. Therefore the orbits can change shape and orienta- if taken to the test particle limit, this results in a conservation tion (on timescales much longer than their orbital periods), but of the z-component of the inner orbit’s angular momentum. We not semimajor axes (SMA).

c 0000 RAS, MNRAS 000, 000–000 Secular Dynamics in Hierarchical Three-Body Systems 3

Figure 2. Geometry of the angular momentum vectors. We show the total angular momentum vector (Gtot), the angular momentum vector of the inner orbit (G1) with inclination i1 with respect to Gtot and the angular momentum vector of the outer orbit (G2) with inclination i2 with respect to Gtot. The angle between G1 and G2 deﬁnes the mutual inclination itot = i1 + i2. The invariable plane is perpendicular to Gtot.

Figure 1. Coordinate system used to describe the hierarchical triple system (not to scale). Here ’c.m.’ denotes the center of mass of the Harrington 1968), inner binary, containing objects of masses m1 and m2. The separation 2 2 k m1m2 k m3(m1 + m2) vector r1 points from m1 to m2; r2 points from ’c.m.’ to m3. The angle H = + (1) 2a1 2a2 between the vectors r1 and r2 is Φ. 2 ∞ j j+1 k j r1 a2 + α Mj Pj (cos Φ) , a2 a1 r2 j=2 X 2 where k is the gravitational constant, Pj are Legendre polyno- mials, Φ is the angle between r1 and r2 (see Figure 1) and

We first define our basic notations. The system consists of j−1 j−1 m1 − (−m2) a close binary (bodies of masses m1 and m2) and a third body Mj = m1m2m3 j . (2) (m1 + m2) (mass m3). It is convenient to describe the orbits using Jacobi coordinates (Murray and Dermott 2000, p. 441-443). Let r1 be Note that we have followed the convention of Harrington (1969) the relative position vector from m1 to m2 and r2 the position and chosen our Hamiltonian to be the negative of the total en- vector of m3 relative to the center of mass of the inner binary (see ergy, so that H > 0 for bound systems. fig. 1). Using this coordinate system the dominant motion of the We adopt the canonical variables known as Delaunay’s el- triple can be divided into two separate Keplerian orbits: the rel- ements, which provide a particularly convenient dynamical de- ative orbit of bodies 1 and 2, and the orbit of body 3 around the scription of our three-body system (e.g. Valtonen and Karttunen center of mass of bodies 1 and 2. The Hamiltonian for the system 2006). The coordinates are chosen to be the mean anomalies, l1 can be decomposed accordingly into two Keplerian Hamiltonians and l2, the longitudes of ascending nodes, h1 and h2, and the plus a coupling term that describes the (weak) interaction be- arguments of periastron, g1 and g2, where subscripts 1, 2 denote tween the two orbits. Let the SMAs of the inner and outer orbits the inner and outer orbits, respectively. Their conjugate momenta be a1 and a2, respectively. Then the coupling term in the com- are plete Hamiltonian can be written as a power series in the ratio m1m2 2 L1 = k (m1 + m2)a1 , (3) of the semi-major axes α = a1/a2 (e.g., Harrington 1968). In a m1 + m2 hierarchical system, by definition, this parameter α is small. p m3(m1 + m2) 2 L2 = k (m1 + m2 + m3)a2 , The complete Hamiltonian expanded in orders of α is (e.g., m1 + m2 + m3 p c 0000 RAS, MNRAS 000, 000–000 4 Naoz et al.

2 2 where the mass parameters are G1 = L1 1 − e1 , G2 = L2 1 − e2 , (4) 2 q q 2 L1 β1 = k m1m2 , (16) and a1 2 2 L2 β2 = k (m1 + m2)m3 (17) H1 = G1 cos i1 ,H2 = G2 cos i2 , (5) a2 and 4 7 7 k (m1 + m2) m3 where e1 (e2) is the inner (outer) orbit eccentricity. Note that β3 = . (18) 16 3 3 G1 and G2 are also the magnitudes of the angular momentum (m1m2) (m1 + m2 + m3) vectors (G1 and G2), and H1 and H2 are the z-components of these vectors. Figure 2 shows the resulting conﬁguration of theses vectors. The following geometric relations between the momenta 3 SECULAR EVOLUTION TO THE follow from the law of cosines: QUADRUPOLE ORDER

2 2 2 In this section, we derive the secular quadrupole–level Hamil- Gtot − G1 − G2 cos itot = , (6) tonian. in Appendix A we develop the complete quadrupole- 2G1G2 2 2 2 level secular approximation and in particular in Appendix A3 Gtot + G1 − G2 H1 = , (7) we present the quadrupole–level equations of motion. The main 2Gtot difference between the derivation shown here (see also Appendix 2 2 2 Gtot + G2 − G1 A) and those of previous studies lies in the “elimination of nodes” H2 = , (8) 2Gtot (e.g., Kozai 1962; Jefferys and Moser 1966). This is related to the transition to a coordinate system with the total angular momen- where Gtot = G1 + G2 is the (conserved) total angular mo- tum along the z-axis, which is known as the invariable plane mentum, and the angle between G1 and G2 defines the mutual (e.g., Murray and Dermott 2000). In this coordinate system (see inclination itot = i1 + i2. From eqs. (7) and (8) we find that Figure 2), the longitudes of the ascending nodes differ by π, i.e., the inclinations i1 and i2 are determined by the orbital angular momenta: h1 − h2 = π . (19)

2 2 2 Conservation of the total angular momentum implies that this Gtot + G1 − G2 cos i1 = , (9) relation holds at all times. Many previous works have exploited 2GtotG1 it to explicitly simplify the Hamiltonian by setting h1 − h2 = π 2 2 2 Gtot + G2 − G1 before deriving the equations of motion. After the substitution, cos i2 = . (10) 2GtotG2 the Hamiltonian is independent of the longitudes of ascending nodes (h1 and h2), and this can lead to the incorrect conclusion In addition to these geometrical relations we also have that that H˙ 1 = H˙ 2 = 0 when the canonical equations of motion are derived. Some previous studies incorrectly concluded that the z- H1 + H2 = Gtot = const . (11) components of the orbital angular momenta are always constant (see also Appendix C). The substitution h1 − h2 = π is incorrect The canonical relations give the equations of motion: at the Hamiltonian level because it unduly restricts variations in the trajectory of the system to those where δh1 = δh2. After dLj ∂H dlj ∂H deriving the equations of motion, however, we can exploit the = , = − , (12) dt ∂lj dt ∂Lj relation h1 − h2 = ∆h = π, which comes from the conservation dGj ∂H dgj ∂H of angular momentum. This considerably simplifies the evolution = , = − , (13) equations. We show (Appendices A3 and B) that one can still dt ∂gj dt ∂Gj use the Hamiltonian with the nodes eliminated found in previ- dHj ∂H dhj ∂H = , = − , (14) ous studies (e.g., Kozai 1962; Harrington 1969) as long as the dt ∂hj dt ∂Hj evolution equations for the inclinations are derived from the to- where j = 1, 2. Note that these canonical relations have the op- tal angular momentum conservation, instead of using the canon- posite sign relative to the usual relations (e.g., Goldstein 1950) ical relations. Of course, the correct evolution equations can also because of the sign convention we have chosen for our Hamilto- be calculated from the correct Hamiltonian (without the nodes nian. Finally we write the Hamiltonian through second order in eliminated), which we derive in this section. α as (e.g., Kozai 1962) We note that there are some other derivations of the secular evolution equations that avoid the elimination of the nodes β1 β2 (Farago and Laskar 2010; Laskar and Boué2010; Mardling 2010; 1 H = 2 + 2 + (15) Katz and Dong 2011), and thus do not suffer from this error . 2L1 2L2 4 2 3 L1 r1 a2 4β3 6 (3 cos 2Φ + 1) , 1 L a1 r2 It is possible to eliminate the nodes as long as one does not 2 c 0000 RAS, MNRAS 000, 000–000 Secular Dynamics in Hierarchical Three-Body Systems 5

The secular Hamiltonian is given by the average over the In the TPQ limit, Gtot ≫ G1, so H˙ 1 = −H˙ 2 ≈ 0, and the z- rapidly-varying l1 and l2 in equation (15) (see Appendix A for component of each orbit’s angular momentum is conserved. Out- more details) side this limit, when G1/Gtot is not negligable, H1 and H2 can- not be constant. Note that the TPQ limit, where G1 ≪ Gtot, is C2 2 H2 = [1 + 3 cos(2i2)] [2 + 3e1][1 + 3 cos(2i1)] (20) equivalent to the limit where i2 ≈ 0 appearing in many previous 8 2 2 2 works. + 30e1 cos(2g1) sin (i1) + 3 cos(2∆h)[10e1 cos(2g1) 2 2 2 × {3 + cos(2i1)} + 4(2 + 3e1) sin(i1) ] sin (i2) 2 2 + 12{2 + 3e1 − 5e1 cos(2g1)} cos(∆h) sin(2i1) sin(2i2) 4 OCTUPOLE-LEVEL EVOLUTION 2 + 120e1 sin(i1) sin(2i2) sin(2g1) sin(∆h) In Appendix B, we derive the secular evolution equations to 2 2 − 120e1 cos(i1) sin (i2) sin(2g1) sin(2∆h) , octupole order. Many previous octupole–order derivations pro- vided correct secular evolution equations for at least some of where the elements, in spite of using the elimination of nodes sub- 4 7 7 4 k (m1 + m2) m3 L1 stitution at the Hamiltonian level (e.g. Harrington 1968, 1969; C2 = 3 3 3 3 . (21) 16 (m1 + m2 + m3) (m1m2) L2G2 Sidlichovsky 1983; Krymolowski and Mazeh 1999; Ford et al. Making the usual (incorrect) substitution ∆h → π (i.e. elim- 2000a; Blaes et al. 2002; Lee and Peale 2003; Thompson 2011). inating the nodes), we get the quadrupole-level Hamiltonian that This is because the evolution equations for e2, g2, g1 and e1 can has appeared in many previous works (see, e.g. Ford et al. 2000a): be found correctly from a Hamiltonian that has had h1 and h2 eliminated by the relation h1 − h2 = π; the partial derivatives 2 2 H2(∆h → π) = C2{ 2 + 3e1 3 cos itot − 1 (22) with respect to the other coordinates and momenta are not af- 2 2 fected by the substitution. The correct evolution of H1 and H2 + 15e1sin itotcos(2 g1)} , can then be derived, not from the canonical relations, but from where we have set i1 + i2 = itot. Because this Hamiltonian is total angular momentum conservation. We discuss in more de- missing the longitudes of ascending nodes (h1 and h2), many tails the comparison between this work and previous analyses in previous studies concluded that the z-components (i.e. vertical §5. components) of the angular momenta of the inner and outer or- The octupole-level terms in the Hamiltonian can become bits (i.e., H1 and H2) are constants. important when the eccentricity of the outer orbit is non-zero, We derive the quadrupole–level equations of motions in Ap- and if α is large enough. We quantify this by considering the ratio pendix A3. In particular, we give the equations of motion of the between the octupole to quadrupole-level coefficients, which is z-component of the angular momentum of the inner and outer orbits derived from the Hamiltonian in Eq. (20). As we show in C3 15 m1 − m2 a1 1 = 2 , (24) C2 4 m1 + m2 a2 1 − e the subsequent sections the evolution of H1,2 produces a qualita- 2 tively different evolutionary route for many astrophysical systems where C3 is the octupole-level coefficient [eq. (B1)] and C2 is the considered in previous works. quadrupole-level coefficient [eq. (21)]. We define In Appendix A4 we show that the quadrupole approximation leads to well-defined minimum and maximum eccentricity m1 − m2 a1 e2 ǫM = 2 , (25) m1 + m2 a2 1 − e and inclination. The eccentricity of the inner orbit and the inner 2 (and mutual) inclination oscillate. In the test-particle limit, our which gives the relative significance of the octupole-level term formalism gives the critical initial mutual inclination angles for in the Hamiltonian. This parameter has three important parts; ◦ ◦ large oscillations of 39.2 6 itot 6 140.8 with nearly-zero initial first the eccentricity of the outer orbit (e2), second, the mass 2 inner eccentricity, in agreement with Kozai (1962). difference of the inner binary (m1 and m2) and the SMA ratio . It is easy to show that H1 and H2 are constant only in In the test particle limit (i.e., m1 ≫ m2) ǫM is reduced to the the TPQ limit without using the explicit equations of motion in octupole coefficient introduced in Lithwick and Naoz (2011) and Appendix A. Because the Hamiltonian in Eq. (20) is independent Katz et al. (2011), of g2, G2 = const at the quadrupole level. Combining this with 2 2 2 a1 e2 the geometric relation in Eq. (7), H1 =(Gtot +G1 −G2)/(2Gtot), ǫ = 2 . (26) a2 1 − e and the constantcy of the total angular momentum, Gtot, we have 2 that We call the octupole-level behavior of a system for which ǫM ≪ 1 ˙ is not satisfied the “eccentric Kozai-Lidov” (EKL) mechanism. ˙ G1G1 H1 = . (23) The octupole terms vanish when e2 = 0. Therefore if one ar- Gtot tificially held e2 = 0, in the test-particle limit the inner body’s orbit would be given by the equations derived by Kozai (1962), i.e. conclude that the conjugate momenta are constant, one example is Lidov and Ziglin (1976) and another is Malige et al. (2002) that after eliminating the nodes introduced a different transformation which 2 Note here that the subscripts “1” and “2” refer to the inner bodies overcame the problem. in m1 and m2, but the subscript “2” refers to the outer body in e2.

c 0000 RAS, MNRAS 000, 000–000 6 Naoz et al. by the test particle quadrupole equations. However, at octupole order the value of e2 evolves in time if the inner body is massive. Furthermore, even if the inner body is massless, if the outer body has e2 > 0 then the inner body’s behavior will also be different than in Kozai’s treatment. For example, Lithwick and Naoz (2011) and Katz et al. (2011) find that the inner orbit can flip orientation (see below) even in the test-particle, octupole limit. The octupole-level effects can change qualitatively the evolution of a system. Compared to the quadrupole-level behavior, the eccentricity of the inner orbit in the EKL mechanism can reach a much higher value. In some cases these excursions to very high eccentricities are accompanied by a “flip” of the orbit with re- ◦ spect to the total angular momentum, i.e. starting with i1 < 90 ◦ the inner orbit can eventually reach i1 > 90 (see Figures 6–9 for examples). Chaotic behavior is also possible at the octupole level (Lithwick and Naoz 2011), but not at the quadrupole–level. Given the large, qualitative changes in behavior moving from quadrupole to octupole order in the Hamiltonian, is it possible that similar changes in the secular evolution may occur at even higher orders? The answer to this question probably lies in the elimination of G2 as an integral of motion at octupole order, leaving only four integrals of motion: the energy of the system, and the three components of the total angular momen- Figure 3. Comparison between a direct integration (using a B-S inte- tum. There are no more integrals of motion to be eliminated, grator) and the octupole-level approximation (see Appendix B). The and thus one might expect no more dramatic changes in the evo- red lines are from the integration of the octupole-level perturbation lution when moving to even higher orders. It is possible to see equations, while the blue lines are from the direct numerical integra- this quantitatively for specific initial conditions through compar- tion of the three-body system. Here the inner binary contains a star isons with direct n-body integrations. We compared our octupole of mass 1 M⊙ and a planet of mass 1 MJ, while the outer object is a equations with direct n-body integrations, using the Mercury brown dwarf of mass 40 MJ. The inner orbit has a1 = 6 AU and the outer orbit has a2 = 100 AU. The initial eccentricities are e1 = 0.001 software package (Chambers and Migliorini 1997). We used both ◦ Burlisch-Stoer and symplectic integrators (Wisdom and Holman and e2 = 0.6 and the initial relative inclination itot = 65 . The thin horizontal line in the top panel marks the 90◦ boundary, separating 1991) and found consistent results between the two. We present prograde and retrograde orbits. The initial mutual inclination of 65◦ the results of a typical integration compared to the integration corresponds to an inner and outer inclination with respect to the total of the octupole-level secular equations in Figure 3. The initial angular momentum (parallel to z) of 64.7◦ and 0.3◦, respectively. Here, ◦ conditions (see caption) for this system are those of Naoz et al. the arguments of pericenter of the inner orbit is set to g1 = 45 and (2011), Figure 1. We find good agreement between the direct in- the outer orbit set to zero initially. The SMA of the two orbits (not tegration and the secular evolution at octupole order. Both show shown) are nearly constant during the direct integration, varying by a beat-like pattern of eccentricity oscillations, suggesting an in- less than 0.02 percent. The agreement in both period and amplitude terference between the quadrupole and octupole terms, and both of oscillation between the direct integration and the octupole-level ap- methods show similar flips of the inner orbit. proximation is quite good.

5 IMPLICATIONS AND COMPARISON WITH object. While the TPQ formalism incorrectly assumes that the PREVIOUS STUDIES orbit of the outer body is fixed in the invariable plane, and therefore the inner body’s vertical angular momentum is constant, the The Kozai (1962) and Lidov (1962) equations of motions are quadrupole-level equations presented in Appendix A3 do not. correct to quadrupole order and for a test particle, but differ We compare the two formalisms in Figure 4. We consider from the correct evolution equations for non-test-particle inner the triple system PSR B1620−26 located near the core of the orbits and/or at octupole order. In this Section we show how globular cluster M4. The inner binary contains a millisecond ra- these differences give rise to qualitatively different evolutionary dio pulsar of m1 = 1.4M⊙ and a companion of m2 = 0.3M⊙ behaviors than those assumed in some previous works. (McKenna and Lyne 1988). Following Ford et al. (2000b), we adopt parameters for the outer perturber of m3 = 0.01 M⊙ and 5.1 Massive Inner Object at the Quadrupole Level e2 = 0. Note that Ford et al. (2000b) found e2 = 0.45, but it is interesting to show that even for an axisymmetric outer po- The danger with working in the wrong limit is apparent if we tential the evolution of the system is qualitatively different then consider an inner object that is more massive then the outer the TPQ approximation (see the caption for a full description

c 0000 RAS, MNRAS 000, 000–000 Secular Dynamics in Hierarchical Three-Body Systems 7 of the initial conditions). Note that the actual measured inner binary eccentricity is e1 ∼ 0.045, however in order to illustrate the difference we adopt a higher value (e1 = 0.5 ). For these initial conditions ǫM = 0.036, so a careful analysis would require incorporating the octupole–order terms in the motion; neverthe- less, we consider the evolution of the system to quadrupole order for comparison with the TPQ formalism. We have verified, however, that the neglected octupole–order effects do not qualitatively change the behavior of the system. This is because the outer companion mass is low, and hence the inner orbit does not exhibit large amplitude oscillations3 . For the comparison, we do not compare the (constant) H1 from the TPQ formalism to the (varying) H1 of the correct formalism. Instead, we compare the (varying) H1 from the correct formalism (solid red line) with G1 cos itot (dashed blue line), which is the vertical angular momentum that would be inferred in our formalism if the outer orbit were instantaneously in the invariable plane, as assumed in the TPQ formalism. In Figure 4, the mutual inclination oscillates between 106.7◦ to 57.5◦, and thus crosses 90◦. These oscillations are mostly due to the oscillations of the outer orbit’s inclination, while i1 does not change by more than ∼ 1◦ in each cycle. Clearly, the outer orbit does not lie in the fixed invariable plane. Figure 4, bottom Figure 4. 2 Comparison between the standard TPQ formalism (dashed panel, shows 1 − e1 cos itot, which, in the TPQ limit, is the blue lines) and our method (solid red lines) for the case of vertical angular momentum of the inner body. p PSR B1620−26. Here the inner binary is a millisecond pulsar of mass We can evaluate analytically the error introduced by the 1.4 M⊙ with a companion of m2 = 0.3M⊙, and the outer body has application of the TPQ formalism to this situation. We com- mass m3 = 0.01M⊙. The inner orbit has a1 = 5 AU and the outer pare the vertical angular momentum (H1) as calculated here to orbit has a2 = 50 AU (Ford et al. 2000b). The initial eccentricities are ◦ T P Q 2 e1 = 0.5 and e2 = 0 and the initial relative inclination itot = 70 . The H1 = L1 1 − e1 cos i = const.. The relative error between T P Q thin horizontal line in the top panel marks the 90◦ boundary, sepa- the formalisms is H1 /H1 − 1. In Figure 5 we show the ratio p rating prograde and retrograde orbits. The initial mutual inclination between the inner orbit’s vertical angular momentum in the TPQ ◦ T P Q of 70 corresponds to an inner and outer inclination with respect to limit (i.e., H = G1 cos i) and equation (A26) as a function 1 the total angular momentum (parallel to ˆz) of 6.75◦ and 63.25◦, re- of the total angular momentum ratio, G1/G2, for various incli- spectively. The argument of pericenter of the inner orbit is initially set nations. Note that this error can be calculated without evolving 120◦, while the outer orbit’s is set to zero. We consider, from top to the system by using angular momentum conservation, Eq. (6). bottom, the mutual inclination itot, the inner orbit’s eccentricity and −4 The TPQ limit is only valid when G1/G2 < 10 . 2 ∼ q1 − e1 cos itot, which the standard formalism assumes to be constant (dashed line). 5.2 Octupole–Level Planetary Dynamics

Recent measurements of the sky-projected angle between the or- therefore have difficulty accounting for the observed retro- bits of several hot Jupiters and the spins of their host stars have grade hot Jupiter orbits. An alternative migration scenario shown that roughly one in four is retrograde (Gaudi and Winn that can account for the retrograde orbits is the secular in- 2007; Triaud et al. 2010; Albrecht et al. 2012). If these plan- teraction between a planet and a binary stellar companion ets migrated in from much larger distances through their (Wu and Murray 2003; Fabrycky and Tremaine 2007; Wu et al. interaction with the protoplanetary disk (Lin and Papaloizou 2007; Takeda et al. 2008; Correia et al. 2011). For an extremely 1986; Masset and Papaloizou 2003), their orbits should have distant and massive companion (ǫM → ǫ ≪ 1) the quadrupole 4 2 low eccentricities and inclinations . Disk migration scenarios test-particle approximation applies, and 1 − e1 cos i1 is nearly constant (where the planet is the massless body). Although this p forbids orbits that are truly retrograde (with respect to the to- 3 Unlike the test particle octupole-level approximation tal angular momentum of the system), if the inner orbit be- (Lithwick and Naoz 2011; Katz et al. 2011), backreaction of the gins highly inclined relative to the outer star’s orbit and aligned outer orbit may suppress the eccentric Kozai effect. We address this in further detail in Teyssandier et al. (in prep). 4 This assumption can be invalid if there are significant magnetic in- (e.g. Thies et al. 2011; Boley et al. 2012) or if there is an episode of teractions between the star and the protoplanetary disk (Lai et al. planet-planet scattering following planet formation (Chatterjee et al. 2010) or if there are interaction with another star in a stellar cluster 2008; Nagasawa et al. 2008) see also Merritt et al. (2009).

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Figure 5. The ratio between the correct, changing z-component of the Figure 6. Evolution of a planetary system with m1 = 1 M⊙, m2 = angular momentum, H1, and the TPQ assumption often used in the 1 MJ and m3 = 2 MJ , with a1 = 4 AU and a2 = 45 AU. We initialize T P Q ◦ ◦ literature, H1 = G1 cos itot. This ratio was calculated analytically the system at t = 0 with e1 = 0.01, e2 = 0.6, g1 = 180 , g2 = 0 and ◦ ◦ ◦ for various total angular momentum ratios, G1/G2, and inclinations. itot = 67 . For these initial conditions i1 = 57.92 and i2 = 9.08 . The curves, from bottom to top, have i = 40, 50, 60, 70, 80 and 89 The z-components of the orbital angular momenta, H1 and H2, are degrees. shown normalized to the total angular momentum of each orbit. The ◦ inner orbit ﬂips repeatedly between prograde (i1 < 90 ) and retrograde ◦ (i1 > 90 ).

with the spin of the inner star, then the star-planet spin-orbit angle can change by more than 90◦ during the secular evolution of the system, producing apparently retrograde orbits (Fabrycky and Tremaine 2007; Correia et al. 2011). Nonetheless, the analysis of triple stars (e.g., Fabrycky and Tremaine 2007; a difficulty with this “stellar Kozai” mechanism is that even with Perets and Fabrycky 2009), see §5.4. < the most optimistic assumptions it can only produce ∼ 10% of A dramatic difference between the octupole and quadrupole– hot Jupiters (Wu et al. 2007). level of approximation is that the former often generates ex- Wu and Murray (2003), Wu et al. (2007), tremely high eccentricities. In real systems, such high eccentric- Fabrycky and Tremaine (2007) and Correia et al. (2011) ities can be suppressed by tides or GR (e.g., Söderhjelm 1984; studied the evolution of a Jupiter-mass planet in stellar binaries Eggleton et al. 1998; Kiseleva et al. 1998; Borkovits et al. 2004). in the TPQ formalism. For example, the case of HD 80606b Flips can also be prevented because they typically occur shortly (Wu and Murray (2003); Fabrycky and Tremaine (2007, Fig. 1) after extreme eccentricities (see Teyssandier et al. in prep.). In and Correia et al. (2011, also Fig. 1)) was considered with an our previous studies that include tides, planetary perturbers typ- outer stellar companion at 1000 AU. However, if the companion ically allow flips to happen, while stellar perturbers mostly sup- is assumed to be eccentric ǫM is not negligible, and the system press them (Naoz et al. 2011, 2012a) But in both cases, tides is more appropriately described with the test particle octupole– quantitatively affect the evolution. level approximation (e.g., Lithwick and Naoz 2011; Katz et al. Naoz et al. (2011) considered planet-planet secular interac- 2011). Furthermore, the statistical distribution for closer stellar tions with tidal interactions as a possible source of retrograde binaries in Wu et al. (2007) and Fabrycky and Tremaine (2007) hot Jupiters. In this situation ǫM is not small, requiring compu- is only valid in the approximation where the outer orbit’s eccen- tation of the octupole-level secular dynamics. In Figures 6 and 7 tricity is zero. In fact, for the systems considered in those studies we show the evolution of a representative configuration (see the ǫM is not negligible and the octupole–level approximation results caption for a full description of the initial conditions). For this in dramatically different behavior as was shown in Naoz et al. configuration, ǫM = 0.083. Flips of the inner orbit are associated (2012a). The same dramatic difference in behavior also exists in with evolution to very high eccentricity (see Figures 6 and 7).

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Figure 7. Zoom-in on part of the evolution of the point-mass planetary Figure 8. Evolution of an asteroid due to Jupiter’s secular gravita- system in Figure 6. In this zoom-in, we can see that flips in the inner tional perturbations (Kozai 1962). We consider m1 = 1 M⊙, m2 → 0 ◦ orbit—i1 crossing 90 —are associated with excursions to very high and m3 =1MJ , with a1 = 2 AU and a2 = 5 AU. We initialize the sys- ◦ ◦ eccentricity. tem at t = 0 with e1 = 0.2, e2 = 0.05, g1 = g2 = 0 and itot = 65 . We show the TPQ evolution (cyan lines) and the EKL evolution (red lines). The thin horizontal dotted line in the top panel marks the 90◦ boundary, separating prograde and retrograde orbits. The inner orbit flips ◦ ◦ 5.3 Octupole–Level Solar System Dynamics periodically between prograde (i1 < 90 ) and retrograde (i1 > 90 ). Kozai (1962) studied the dynamical evolution of an asteroid due We also show the result of an N-body simulation (blue lines). The thin horizontal dotted line in the bottom panel marks the eccentricity to Jupiter’s secular perturbations. He assumed that Jupiter’s corresponding to a collision with the solar surface, 1 − e1 = R⊙/a1. eccentricity is strictly zero. However, Jupiter’s eccentricity is ∼ 0.05, and thus studying the evolution of a test particle in the asteroid belt (a1 ∼ 2 − 3 AU) places the evolution in a regime oid approaches about 1 AU from Jupiter’s orbit. To determine where the eccentric Kozai-Lidov effect could be significant, with the importance of these effects, we ran an N-body simulation ǫM = ǫ = 0.03 (Lithwick and Naoz 2011; Katz et al. 2011). using the Mercury software package (Chambers and Migliorini We considered the evolution of asteroid at 2 AU (assumed 1997). We used both Bulirsch-Stoer and symplectic integrators to be a test particle) due to Jupiter at 5 AU with eccentric- (Wisdom and Holman 1991). The results are depicted at Figure ity of e2 = 0.05 (see the caption for a full description of the 8, which show that the TPQ limit is indeed inadequate for the initial conditions). The asteroid is a test particle and therefore system. In addition the octupole–level approximation has some i1 ≈ itot. In Figure 8 we compare the evolution of an asteroid deviations from the direct N-body integration, particularly in the using the TPQ limit (e.g., Kozai 1962; Thomas and Morbidelli high eccentricity regime. Note that the evolution of the asteroid 1996; Kinoshita and Nakai 2007) and the octupole-level evolu- in the direct integration resulted in a collision with the Sun5. In tion discussed here. For this value of ǫ, the eccentric Kozai-Lidov reality, it is likely that a planetary encounter would remove the effect significantly alters the evolution of the asteroid, even driv- asteroid from the solar system before this point. In contrast to ing it to such high inclination that the orbit becomes retrograde. the EKL mechanism, assuming zero eccentricity for Jupiter re- Though we deal only with point masses in this work, note that sults in consistent results between the secular evolution and the the eccentricity is so high that the inner orbit’s pericenter lies direct integration (Thomas and Morbidelli 1996). well within the sun. As shown in Figure 8, taking into account Jupiter’s ec- The value of ǫ here is mainly due to the relative high α in centricity (∼ 0.05), produces a dramatically different evolu- the problem (an issue raised in the original work on this problem tionary behavior, including retrograde orbits for the asteroid. (Kozai 1962)). The system is very packed which raises questions with regards to the validity of the hierarchical approximation. Even in the EKL formalism, such high eccentricities occur that 5 As noted in Lithwick and Naoz (2011) for very small periapse the the asteriod collides with the sun and the apo-center of the aster- integration becomes extremely costly.

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Thomas and Morbidelli (1996) applied the TPQ formalism to the asteroid-Jupiter setting (see for example their Figure 2 for a1 = 3 AU). Kinoshita and Nakai (2007) developed an analytical solution for the TPQ limit (see also Kinoshita and Nakai 1991, 1999). The TPQ formalism has also been applied to the study of the outer solar system. Kinoshita and Nakai (2007) applied their analytical solution to Neptune’s outer satellite Laomedeia. This system has ǫ → 0 and thus the TPQ limit there is justiﬁed. In addition, Perets and Naoz (2009) have studied the evolution of binary minor planets using the TPQ approximation. In this problem ǫ → 0 and thus the TPQ approximation is valid. Lidov and Ziglin (1976, sections 3–4) also solved analytically the quadrapole–level approximation but, unlike Kinoshita and Nakai (2007), they did not restrict themselves to the TPQ limit, and used the total angular momentum conservation law in order to calculate the inclinations. Thus, their formalism is equivalent to ours at quadrupole–order. Later, Mazeh and Shaham (1979) also derived evolution equations outside the TPQ limit (their eqs. A1-A8), allowing for small eccentricities and inclinations of the outer body.

Figure 9. An example of dramatically different evolution between 5.4 Octupole–Level Perturbations in Triple Stars the quadruple and octupole approximations for a triple-star system. The system has m1 = 1M⊙, m2 = 0.25M⊙ and m3 = 0.6M⊙, The evolution of triple stars has been studied by with a1 = 60 AU and a2 = 800 AU. We initialize the system with ◦ ◦ many authors using the standard (TPQ) formal- e1 = 0.01, e2 = 0.6, g1 = g2 = 0 and itot = 98 , taken from ◦ ism (e.g., Mazeh and Shaham 1979; Eggleton et al. Fabrycky and Tremaine (2007). For these initial conditions i1 = 90.02 ◦ 1998; Kiseleva et al. 1998; Mikkola and Tanikawa 1998; and i2 = 7.98 . We show both the (correct) quadrupole-level evolution Eggleton and Kiseleva-Eggleton 2001; Fabrycky and Tremaine (light-blue lines) and the octupole-level evolution (red lines). H1 and 2007; Perets and Fabrycky 2009). In some cases the corrected H2, the z-components of the angular momenta of the orbits, are normalized to the total angular momentum. Note that the octupole-level formalism derived here can give rise to qualitatively different evolution produces periodic transitions from prograde to retrograde results. We show that some of the previous studies should be inner orbits (relative to the total angular momentum), while at the repeated in order to account for the correct dynamical evolution, quadrupole-level the inner orbit remains prograde. See Figure 10 for and give one example where the eccentric Kozi-Lidov mechanism comparison with direct numerical integration of the three-body system. dramatically changes the evolution. Fabrycky and Tremaine (2007) studied the distribution of triple star properties using Monte Carlo simulations. We choose a particular system from their triple-star suite of simulations to in reality flips in similar systems may be suppressed. Similarly illustrate how the dynamics including the octupole order can be the high eccentricities often excited through the eccentric Kozai qualitatively different from what would be seen at quadrupole mechanism can also lead to compact object binary merger. order (see the caption for a full description of the initial con- The possibility of forming blue stragglers through secu- ditions). For this system ǫM = 0.042 (and ǫ = 0.0703). The lar interactions in triple star systems has been suggested by evolution of the system is shown in Figure 9. At octupole or- Perets and Fabrycky (2009) and Geller et al. (2011). As shown der, the inclination of the inner orbit oscillates between about in Krymolowski and Mazeh (1999); Ford et al. (2000a) and in 40◦ and 140◦, often becoming retrograde (relative to the to- the example above the minimum pericenter distance of the in- tal angular momentum), while the quadrupole–order behavior ner binary can differ significantly between the TPQ and EKL is very different and the inner orbit remains always prograde. formalisms. This suggests that using the correct EKL formal- The octupole–order treatment also gives rise to much higher ec- ism could significantly increase the computed likelihood of such centricities (Krymolowski and Mazeh 1999; Ford et al. 2000a). In a formation mechanism for blue stragglers. Figure 10 we compare the octupole–level evolution (of the same For many years CH Cygni was considered to be an interest- system) with direct 3–body integration. ing triple candidate because it exhibits two clear distinguish- The evolution shown in Figure 9 is for point-mass stars; in able periods (e.g. Donnison and Mikulskis 1995; Skopal et al. reality, these high-eccentricity excursions would actually drive 1998; Mikkola and Tanikawa 1998; Hinkle et al. 1993). However, the inner binary to its Roche limit, leading to mass transfer. For a triple system model based on the TPQ Kozai mechanism these high eccentricities tides will play an important role and thus (Mikkola and Tanikawa 1998) did not reproduce the observed

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Figure 10. The evolution for the first 6 Myr, of Figure 9 where we Figure 11. An example of dramatically different evolution between show a comparison between direct 3–body integration (using a B-S the quadruple and octupole approximations for a triple star system integrator), and the octupole–level of approximation. The red lines representing the best-fit parameters from the Mikkola and Tanikawa are from the integration of the octupole-level perturbation equations, (1998) analysis of CH Cygni. The system has m1 = 3.51M⊙, m2 = while the black lines are from the direct numerical integration of the 0.5M⊙ and m3 = 0.909M⊙, with a1 = 0.05 AU and a2 = 0.21 AU. We ◦ ◦ three-body system. initialize the system with e1 = 0.32, e2 = 0.6, g1 = 145 , g2 = 0 and ◦ ◦ ◦ itot = 72 . For these initial conditions i1 = 57.02 and i2 = 14.98 . We show both the (non-TPQ) quadrupole-level evolution (light-blue lines) and the octupole-level evolution (red lines). H1 and H2, the z- masses of the system (Hinkle et al. 1993, 2009). Applying the cor- components of the angular momenta of the orbits, are normalized to rected formalism in this paper to the system parameters derived the total angular momentum. Note that the octupole-level evolution in Mikkola and Tanikawa (1998) gives a very different evolution produces periodic transitions from prograde to retrograde inner orbits than in the TPQ formalism6. Therefore, it seems likely that an (relative to the total angular momentum), while at the quadrupole- analysis based on the formalism discussed in this paper would level the inner orbit remains prograde. To avoid clutter in the figure we give a significantly different fit. In Figure 11 we illustrate the have omitted the TPQ result. In the TPQ formalism, the evolution of differences between the TPQ, correct quadrupole, and octupole the inclination and eccentricity are similar to the general quadrupole- evolution of the system. The best-fit parameters of the system level approximation, but H1,2 are constant. are taken from Mikkola and Tanikawa (1998) where ǫM = 0.14 (see the caption for a full description of the initial conditions, where we allowed for a freedom in our choice of e2,g1,g2 and itot since the best fit was found using the TPQ limit, at which parable masses for the inner orbit, and low eccentricity of e2 is fixed). Note that the choice of the inner eccentricity does the outer orbit (i.e., ǫM << 1). Kiseleva et al. (1998) and not strongly influence the evolution while the choice of the outer Eggleton and Kiseleva-Eggleton (2001) studied the Algol triple orbit’s eccentricity does. Most importantly, the rather large ǫM system (Lestrade et al. 1993) using the TPQ equations. The for this system implies that the system is not stable, i.e., the TPQ equations were also used in the paper that introduced averaging over the orbits is not justified. From direct integration the influential KCTF mechanism (Mazeh and Shaham 1979; we found that the system undergoes strong encounters and the Eggleton et al. 1998). Note that tides dominate the evolution of inner binary collides in this example. the Algol system today (e.g., Söderhjelm 2006). Figure 12 com- It is also interesting to investigate a system for which pares the evolution computed in the (incorrect) TPQ formal- the eccentric Kozai mechanism is suppressed due to com- ism, the correct quadrupole formalism, and the octupole-level EKL formalism applied to an Algol–like system. The correct quadrupole formalism decreases the minimum value of 1 − e1 6 Mikkola and Tanikawa (1998) also found somewhat different set of by almost a factor of 2 relative to the TPQ formalism. The re- parameters when producing a fit for data set with less weight for the duced pericenter distance would strongly increase the effects of data of 1983 due to large noise in the active phase of the system. tidal friction (not included here), which may lead to rapid cir-

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Figure 12. The time evolution of an Algol–like system (Eggleton et al. Figure 13. The time evolution of Algol–like system using the or- 1998), with (m1, m2, m3) = (2.5, 2, 1.7) M⊙. The inner orbit has bital parameters taken from Baron et al. (2012), with (m1, m2, m3)= a1 = 0.095 AU and the outer orbit has a2 = 2.777 AU. The initial (3.17, 0.7, 1.7) M⊙. The inner orbit has a1 = 0.062 AU and the outer eccentricities are e1 = 0.01 and e2 = 0.23 and the initial relative incli- orbit has a2 = 2.68 AU. The initial eccentricities are e1 = 0.001 and ◦ ◦ nation itot = 100 . The z-components of the inner and outer orbital e2 = 0.23 and the initial relative inclination itot = 90 . The initial ◦ angular momentum, H1 and H2 are normalized to the total angu- mutual inclination of 90 corresponds to inner- and outer-orbit incli- lar momentum. The initial mutual inclination of 100◦ corresponds to nations of 86.4◦ and 3.6◦, respectively. We consider only the octupole- inner- and outer-orbit inclinations of 91.6◦ and 8.4◦, respectively. We level evolution. Compare this to the evolution in Figure 12. consider the (correct) quadrupole-level evolution (blue lines), octupole- level evolution (dashed lines) and also the standard (incorrect) TPQ evolution. In the latter we have assumed, as in previous papers, that of the correct dynamical evolution. It would be interesting to itot = i1, which results in the discrepancy between the inclination study stellar evolution including tides in the context of the EKL values. See also Figure 13 for the evolution of the Algol–like system using the updated masses and orbital parameter, following Baron et al. mechanism, for a system such as Algol. (2012). 5.5 The Danger of the Quadrupole–Level of Approximation cularization of the inner orbit. The octupole-level computation The octupole-level Hamiltonian and equations of motion were decreases the minimum pericenter distance by a further 40%. previously derived by Harrington (1968, 1969); Sidlichovsky Note that the masses and orbital parameters used (1983); Marchal (1990); Krymolowski and Mazeh (1999); in Kiseleva et al. (1998) and Eggleton and Kiseleva-Eggleton Ford et al. (2000a); Blaes et al. (2002) and Lee and Peale (2001) are out of date. New observations (e.g., Baron et al. 2012) (2003). Most of the equations of motion can be derived correctly find the secondary mass to be smaller then the primary and the when applying the elimination of the nodes—only the H˙ 1 and ◦ mutual inclination to be closer to 90 . In Figure 13 we show the H˙ 2 equations are affected. These authors calculated the time octupole–level evolution of the system considering the new pa- evolution of the inclinations (i.e. H1 and H2) from the total rameters. In the absence of any additional physical mechanism, (conserved) angular momentum, and thus avoided the problem such as general relativity, tides, mass transfer, etc., the EKL that arises when eliminating the nodes from the Hamiltonian. mechanism could play a very important role in the dynamical In appendix B we show the complete set of equations of motion evolution of the system. for the octupole-level approximation, derived from a correct We note that the inner binary in the Algol system is dom- Hamiltonian, including the nodal terms. inated by tidal effects (Söderhjelm 1975; Kiseleva et al. 1998; As displayed here the octupole-level approximation gives Eggleton and Kiseleva-Eggleton 2001) and figures 12 and 13 do rise to a qualitatively different evolutionary behavior for not represent the system today but an Algol–like analogy. We use cases where ǫM [see eq. (25)] is not negligible. We note the Algol parameters here only to show hypothetical outcomes that many previous studies applied the quadrupole-level

c 0000 RAS, MNRAS 000, 000–000 Secular Dynamics in Hierarchical Three-Body Systems 13 approximation, which may lead to significantly different particle limit these values converge to the well-known critical in- ◦ ◦ results (e.g., Mazeh and Shaham 1979; Quinn et al. 1990; clinations (39.2 6 i0 6 140.8 ) for large oscillatory amplitudes. Bailey et al. 1992; Innanen et al. 1997; Eggleton et al. 1998; The most notable outcome of the results presented here hap- Mikkola and Tanikawa 1998; Eggleton and Kiseleva-Eggleton pens in the octupole-level of approximation (which we call the 2001; Valtonen and Karttunen 2006; Fabrycky and Tremaine EKL formalism), when the inner orbit flips from prograde to ret- 2007; Wu et al. 2007; Zdziarski et al. 2007; Takeda et al. 2008; rograde with respect to the total angular momentum. Just before Perets and Fabrycky 2009). Neglecting the octupole-level ap- the flip the inner orbit has an excursion of extremely high eccen- proximation can cause changes in the dynamics varying from tricity. In the presence of tidal forces (not included in this study) a few percent to completely different qualitative behavior. the outcome of a system can be different than the one assumed Some other derivations of octupole–order equations of mo- while using the TPQ formalism. Krymolowski and Mazeh (1999), tion dealt with the secular dynamics in a general way, without us- Ford et al. (2000a), Blaes et al. (2002), Lee and Peale (2003) and ing Hamiltonian perturbation theory or elimination of the nodes Laskar and Boué(2010) present the correct octupole equations (Farago and Laskar 2010; Laskar and Boué2010; Mardling 2010; of motion. Had these authors integrated their equations for sys- Katz and Dong 2011). In these works there were no references to tems such as those presented in this paper, they could already the discrepancy between these derivations and the previous stud- have discovered the possibility of flipping the inner orbit. ies. Also, note that the results of Holman et al. (1997) are based on a direct N-body integration, and thus are not subject to the errors mentioned above. ACKNOWLEDGMENTS We thank Boaz Katz, Rosemary Mardling and Eugene Chiang for useful discussions. We thank Staffan Söderhjelm, our referee for 6 CONCLUSIONS very useful comments the improved the manuscript in great deal. We have shown that the “standard” TPQ Kozai formalism We also thank Keren Sharon and Paul Kiel for comments on the (Kozai 1962; Lidov 1962) has been applied in inappropriate situ- manuscript. S.N. supported by NASA through a Einstein Post- ations. A common error in the implementation of the relevant doctoral Fellowship awarded by the Chandra X-ray Center, which Hamiltonian mechanics (premature elimination of the nodes) is op- erated by the Smithsonian Astrophysical Observatory for leads to the (incorrect) conclusion that the conservation of the NASA under contract PF2-130096. Y.L. acknowledges support z-component of each orbit’s angular momentum from the TPQ from NSF grant AST-1109776. Simulations for this project were dynamics generalizes beyond the TPQ approximation. Correct- performed on the HPC cluster fugu funded by an NSF MRI ing the formalism we find that the z-components of both the inner award. and outer orbits’ angular momenta in general change with time at both the quadrupole and octupole level. The conservation of the inner orbit’s z-component of the angular momentum (the famous 2 REFERENCES 1 − e1 cos i = constant) only holds in the quadrupole-level test particle approximation. 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for ”Prompt” Type Ia Supernovae, Gamma-Ray Bursts, and coordinate system is given by (see Murray and Dermott 2000, Other Exotica. ApJ , 741, 82. chapter 2.8, and Figure 2.14 for more details) Tokovinin, A. A. (1997). On the multiplicity of spectroscopic r1 inv 1 1 1 r binary stars. Astronomy Letters, 23, 727–730. , = Rz(h )Rx(i )Rz(g ) 1,orb , (A1) Triaud, A. H. M. J., Collier Cameron, A., Queloz, D., An- where the subscript “inv” and “orb” refer to the invariable and derson, D. R., Gillon, M., Hebb, L., Hellier, C., Loeillet, B., orbital coordinate systems, respectively. The rotation matrices Maxted, P. F. L., Mayor, M., Pepe, F., Pollacco, D., Ségransan, Rz and Rx as a function of rotation angle, θ, are D., Smalley, B., Udry, S., West, R. G., and Wheatley, P. J. (2010). Spin-orbit angle measurements for six southern tran- cos θ − sin θ 0 Rz(θ)= sin θ cos θ 0 (A2) siting planets. New insights into the dynamical origins of hot   Jupiters. Aap, 524, A25+. 0 0 1   Valtonen, M. and Karttunen, H. (2006). The Three-Body Prob- and lem. Vashkov’yak, M. A. (1999). Evolution of the orbits of distant 1 0 0 Rx(θ)= 0 cos θ − sin θ . (A3) satellites of Uranus. Astronomy Letters, 25, 476–481.   Veras, D. and Ford, E. B. (2010). Secular Orbital Dynamics of 0 sin θ cos θ   Hierarchical Two-planet Systems. ApJ , 715, 803–822. Thus, the angle between r1 and r2 is given by: Wen, L. (2003). On the Eccentricity Distribution of Coalesc- T −1 −1 −1 ing Black Hole Binaries Driven by the Kozai Mechanism in cos Φ = ˆr2,orbRz (g2)Rx (i2)Rz (h2)Rz(h1)Rx(i1)Rz(g1)ˆr1,orb, Globular Clusters. ApJ , 598, 419–430. (A4) Wisdom, J. and Holman, M. (1991). Symplectic maps for the where ˆr1,2,orb are unit vectors that point along r1,2,orb. In the n-body problem. AJ , 102, 1528–1538. orbital coordinate system, we have Wu, Y. and Murray, N. (2003). Planet Migration and Binary cos (f1,2) Companions: The Case of HD 80606b. ApJ , 589, 605–614. ˆr1,2,orb = sin (f1,2) , (A5)   Wu, Y., Murray, N. W., and Ramsahai, J. M. (2007). Hot 0 Jupiters in Binary Star Systems. ApJ , 670, 820–825.   Zdziarski, A. A., Wen, L., and Gierliński, M. (2007). The su- where f1 (f2) is the true anomaly for the inner (outer) orbit. −1 perorbital variability and triple nature of the X-ray source 4U Note that Rz (h2)Rz(h1) = Rz(h1 − h2) ≡ Rz(∆h), so the 1820-303. MNRAS, 377, 1006–1016. Hamiltonian will depend on the difference in the longitudes of the ascending nodes; in a similar manner, the Hamiltonian depends on f1 and f2 only through expressions of the form f1 + g1 and f2 + g2. Replacing cos Φ in the Hamiltonian, eq. (15), we APPENDIX A: THE QUADRUPOLE LEVEL OF can now integrate over the the mean anomaly angles using the APPROXIMATION Kepler relations between the mean and true anomalies: 2 We develop the complete quadrupole-level secular approxima- 1 ri dli = dfi , (A6) tion in this section. As mentioned, the main difference between 2 1 − ei ai the derivation shown here and those of previous studies lies in the “elimination of nodes” (e.g., Kozai 1962; Jefferys and Moser where for the outer orbitp one should simply replace the subscript 1966), which relates to the transition the invariable plane (e.g., “1” with “2”. Murray and Dermott 2000) coordinate system, where the total angular momentum lies along the z-axis. A2 Transformation to Eliminate Mean Motions Because we are interested in the long-term dynamics of the triple A1 Transformation to the Invariable Plane system, we now describe the transformation that eliminates the We choose to work in a coordinate system where the total initial short-period terms in the Hamiltonian that depend of l1 and l2. angular momentum of the system lies along the z axis (see Fig- The technique we will use is known as the Von Zeipel transfor- ure 2),; the x-y plane in this coordinate system is known as the mation (for more details, see Brouwer 1959). invariable plane (e.g., Murray and Dermott 2000), and therefore Write the triple-system Hamiltonian in eq. (15) as we call this coordinate system the invariable coordinate system. K K H = H1 + H2 + H2, (A7) We begin by expressing the vectors r1 and r2 each in a coordi- K K nate system where the periapse of the orbit is aligned with the where H1 and H2 are the Kepler Hamiltonians that describe x-axis and the orbit lies in the x-y plane, called the “orbital co- the inner and outer elliptical orbits in the triple system and H2 ordinate system,” and then rotating each vector to the invariable describes the quadrupole interaction between the orbits. Note 2 coordinate system. The rotation that takes the position vector that H2 is O α , and is the only term in H that depends on l1 in the orbital coordinate system to the position in the invariable or l2. We seek a canonical transformation that can eliminate the c 0000 RAS, MNRAS 000, 000–000 Secular Dynamics in Hierarchical Three-Body Systems 17

K K l1 and l2 terms from H2. Such a transformation must be close to Now let ∂H1 /∂L1 ≡ ω1(L1), and ∂H2 /∂L2 ≡ ω2(L2). Suppose the identity, since H2 ≪ H; let the generating function be that S2 is periodic in l1 and l2 (which are equivalent, at lowest ∗ ∗ 2 order, to l1 and l2 ). Then ∗ ∗ ∗ ∗ ∗ ∗ S(Lj , Gj ,Hj ,lj ,gj , hj ) = Lj lj + Gj gj + Hj hj (A8) ∞ ∞ ∗ ∗ j=1 2 2 −ik1l1 −ik2l2 2 −ik1l1−ik2l2 X α h0+α hk1k2 e +α ω1 −ik1sk1k2 e 2 ∗ ∗ ∗ + α S2(L , G ,H ,lj ,gj , hj ) , k1,k2=1 k1,k2=1 j j j X X ∞ where we indicate the new momenta with a superscript asterix, 2 −ik1l1−ik2l2 ∗ ∗ ∗ + α ω2 −ik2sk1k2 e = H2 (qi ,pi ) , (A19) 2 and S is the non-identity piece of the transformation that we k1,k2=1 will use to eliminate H2. The relationship between the new and X old canonical variables is where ∞ ∂S ∗ 2 ∂S2 −ik1l1−ik2l2 pi = = pi + α (A9) S2 = s0 + sk1k2 e . (A20) ∂qi ∂qi k1,k2=1 X and ∗ The terms dependent on l1 will be eliminated from H2 if ∗ ∂S 2 ∂S2 qi = = qi + α , (A10) ∂p∗ ∂p∗ hk1k2 i i sk1k2 = −i . (A21) ω1k1 + ω2k2 where the momenta pi ∈ {Li, Gi,Hi}, and the coordinates qi ∈ Assuming than the system is far from resonance (that is, that {li,gi, hi}. Because our generating function is time-independent, the new and old Hamiltonians agree when evaluated at the cor- ω1k1 + ω2k2 6= 0 for all k1 and k2), this gives us the necessary S2 responding points in phase space: to eliminate all terms in H2 that depend on l1 or l2, leaving 2π ∗ ∗ ∗ ∗ ∗ ∗ 2 1 ∗ ∗ ∗ ∗ H(qi,pi)= H (qi ,pi ) (A11) H2 (qi ,pi )= α h0 = 2 dl1 dl2 H2 (qi ,pi ) . (A22) 4π 0 when the phase space coordinates satisfy equations (A9) and Z (A10). Inserting these relations into the un-transformed Hamil- That is, our canonical transformation to eliminate the rapidly- tonian, and expanding to lowest order in α2, we have oscillating parts of H has left us with a Hamiltonian that is the average over the oscillation period of the original Hamiltonian7. ∗ ∗ 2 ∂H ∂S2 2 ∂H ∂S2 ∗ ∗ ∗ H(qi ,pi )+ α − α ∗ = H (qi ,pi ) . (A12) The value of the Hamiltonian in equation (15) averaged over ∂pi ∂qi ∂qi ∂pi the mean motions is

Equating terms order-by-order in α gives ∗ C2 2 2 1 K ∗ ∗ ∗K ∗ ∗ H2 = {[1 + 3 cos(2i )] [2 + 3e1][1 + 3 cos(2i )] (A23) H1 (qi ,pi )= H1 (qi ,pi ), (A13) 8 2 2 2 + 30e1 cos(2g1) sin (i1) + 3 cos(2∆h)[10e1 cos(2g1) K ∗ ∗ ∗K ∗ ∗ H2 (q ,p )= H2 (q ,p ), (A14) 2 2 2 i i i i × (3 + cos(2i1))+4(2+3 e1) sin(i1) ] sin (i2) 2 2 and + 12(2 + 3e1 − 5e1 cos(2g1)) cos(∆h) sin(2i1) sin(2i2) 2 2 2 1 2 1 ∗ ∗ 2 ∂H ∂S2 2 ∂H ∂S2 ∗ ∗ ∗ + 120e1 sin(i ) sin(2i ) sin(2g ) sin(∆h) H2 (qi ,pi )+ α − α ∗ = H2 (qi ,pi ) . 2 2 ∂pi ∂qi ∂qi ∂p − 120e1 cos(i1) sin (i2) sin(2g1) sin(2∆h)} , i=1 i=1 i X X (A15) where C2 was deﬁed in equation (21). Since the last two terms on the left-hand side of this latter equa- 2 K K tion are already O α , only the H1 and H2 parts of H contribute. These Kepler Hamiltonians only depend on L1 and L2, A3 The Quadrupole–level Equations of Motion 2 so there are only two non-zero partials of H at order α : We use the canonical relations [equations (12)] in order to derive K K ∗ ∗ 2 ∂H1 ∂S2 2 ∂H2 ∂S2 ∗ ∗ ∗ the equations of motion from the Hamiltonian. In our treatment, H2 (qi ,pi )+ α + α = H2 (qi ,pi ) . (A16) ∂L1 ∂l1 ∂L2 ∂l2 both H1 and H2 evolve with time because the Hamiltonian is not

We must use the terms that depend on S2 to cancel any terms independent of h1 and h2. From eq. (7), we see that ∗ ∗ ∗ in H2 that depend on l1 and l2 . Note that H2 is periodic in l1 G1 G2 ∗ ˙ ˙ ˙ and l with period 2π (see equations (A4) and (A5)), so we can H1 = G1 − G2 , (A24) 2 Gtot Gtot write ∞ ∗ ∗ ∗ ∗ 2 2 −ik1l1 −ik2l2 7 H2 (qi ,pi )= α h0 + α hk1k2 e , (A17) Note that the canonical variables are also transformed. They differ 2 k1,k2=1 from the original variables at O α . However, this difference is irrel- X evant when evaluating the interaction between the orbits described by with 2 H2, as this interaction is already O α , and so the differences be- 2π ∗ ∗ 1 ∗ ∗ ∗ ∗ ik1l1 +ik2l2 tween the original and transformed variables contribute at sub-leading hk1k2 = dl dl H2 (q ,p ) e . (A18) 4π2α2 1 2 i i order. Z0 c 0000 RAS, MNRAS 000, 000–000 18 Naoz et al. and from eq. (11) we see that H˙ 1 = −H˙ 2. The quadrupole-level Another useful parameter is the inclination, which can be found Hamiltonian does not depend on g2; thus the magnitude of the through the z-component of the angular momentum: 8 outer orbit’s angular momentum, G2, is constant , and therefore d(cos i1) H˙ 1 G˙ 1 ˙ = − cos i1 , (A35) G1G1 dt G1 G1 H˙ 1 = . (A25) Gtot and similarly for i2 (but note again that G˙ 2 = 0 to quadrupole From relations (12-14) we have H˙ 1 = ∂H/∂h1, and G˙ 1 = order). ∂H/∂g1. The former gives ˙ 2 H1 = −30C2e1 sin i2 sin itot sin(2g1) . (A26) A4 Maximum Eccentricity and “Kozai” Angles in and the latter evaluates to the Quadrupole Approximation 2 2 G˙ 1 = −30C2e1 sin itot sin(2g1) . (A27) First note that settinge ˙1 = 0 also means that G˙ 1 = 0. The values of the argument of periapsis that satisfy these relations are: g1 = Employing the law of sines, Gtot/ sin itot = G1/ sin i2 = 0+ πn/2, where n = 0, 1, 2... . Also, setting G˙ 1(e1,max,min) = 0 G2/ sin i1, equation (A26) can also be written as means that H˙ 1(e1,max,min) =0 and i˙1 = 0, i.e., an extremum of G1 2 2 the eccentricity is also an extremum of both the inner and outer H˙ 1 = − 30C2e1 sin itot sin(2g1) , (A28) Gtot inclinations. which satisfies the relation in eq. (A25). The evolution of the The conservation of the total angular momentum, i.e., G1 + arguments of periapse are given by G2 = Gtot sets the relation between the total inclination and inner orbit eccentricity. We re-write equation (6) as 1 2 g˙1 = 6C2 [4 cos itot + (5 cos(2g1) − 1) (A29) G1 2 2 2 2 2 2 L1(1−e1)+2L1L2 1 − e1 1 − e2 cos itot = Gtot −G2 , (A36) 2 2 cos itot 2 q q × (1 − e1 − cos itot)] + [2 + e1(3 − 5 cos(2g1))] , where in the quadrupole-level approximation e2 and G2 are con- G2 stant. The right hand side of the above equation is set by the and initial conditions. In addition, L1, and L2 [see eqs. (3) and (4)] 2 cos itot 2 are also set by the initial conditions. Using the conservation of g˙2 = 3C2 [2 + e1(3 − 5 cos(2g1))] (A30) G1 energy we can write, for the minimum eccentricity case (i.e., setting g1 = 0) 1 2 2 2 + [4 + 6e1 + (5 cos itot − 3)(2 + e1[3 − 5 cos(2g1)])] . G2 E 2 2 2 = 3 cos itot(1 − e1) − 1 + 6e1 , (A37) Previous quadrupole-level calculations that made the substitu- 2C2 tion error in the Hamiltonian lack the 1/G2 terms in these equa- where we also used the relation ∆h = π. We find a similar equations. The evolution of the longitudes of ascending nodes is given tion if we set g1 = π/2: by E 2 2 2 = 3 cos itot(1+4e1) − 1 − 9e1 . (A38) 3C2 2 2 2C2 h˙ 1 = − {2 + 3e1 − 5e1 cos (2g1)} sin (2itot) (A31) G1 sin i1 Equations (A36), (A37) and (A38) give a simple relation between and the total inclination and the inner eccentricity. The remainder of 3C2 2 2 the parameters in the equations are defined by the initial con- h˙ 2 = − {2 + 3e1 − 5e1 cos (2g1)} sin (2itot) . (A32) G2 sin i2 ditions. Thus, using equations (A37) and (A36) we can find the Using the law of sines, G1 sin i1 = G2 sin i2, from which we get minimum eccentricity reached during the oscillation and using h˙ 1 = h˙ 2, as required by the relation h1 −h2 = π. In many systems equations (A38) and (A36) we can find also the maximum and it is useful to calculate the time evolution of the eccentricity, the minimum inclinations. The following example illustrates the obtained through the following relation: relation defined by these equations between the inclination and the eccentricity. dej ∂ej ∂H 0 0 0 = , (A33) For simplicity we set initially e1 = 0, g1 and e2 = 0 (the su- dt ∂Gj ∂gj perscript 0 stand for initial values). In this appendix we consider In the quadrupole approximatione ˙2 = G˙ 2 = 0 (which is not only the quadrupole-level approximation, and thus e2 doesn’t the case at higher order in α; see Appendix B). The eccentricity change. Using these initial conditions (and for some initial mu- evolution for the inner orbit is given by tual inclination i0) we can write equation (A36) as 2 1 − e1 2 2 L1 2 e˙1 = C2 30e1 sin itot sin(2g1) . (A34) 1 − e1 cos itot = cos i0 + e1 . (A39) G1 2L2 q We show these curves for different i0 in Figure A1 (short dashed 8 This conserved quantity is lost at higher orders of the approxima- curves) for a hypothetical system with the parameters of an tion; see §4 and Appendix B. Algol–like system (but with e2 = 0, see §5.4). Note that there is

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the maximum eccentricity. After some algebra we ﬁnd:

2 2 L1 4 L1 L1 2 e1 + 3 + 4 cos i0 + e1 L2 L2 2L2 ! L1 2 + cos i0 − 3 + 5 cos i0 = 0 . (A42) L2

As we approach the TPQ limit, L2 ≫ L1, and this equation becomes

2 5 2 e = 1 − cos i0 , (A43) 1 3 which gives the maximum eccentricity as a function of mutual initial inclination with zero initial inner eccentricity. In Figure A1 we show that this approximation still holds fairly well even for an Algol–like system, where L1/L2 ∼ 0.07. Equation (A43) has been found previously (e.g. Innanen et al. 1997; Kinoshita and Nakai 1999; Valtonen and Karttunen 2006) in the TPQ approximation, but in these works it is assumed valid outside that limit. A solution exists only if the right hand side of this equations is positive, thus we find the critical angles for large Kozai oscillation in the TPQ limit: ◦ ◦ 39.2 6 i0 6 140.8 . (A44) Figure A1. The total inclination and eccentricity relation for an Algol–like system. We show constant energy curves (solid curves, For larger L1/L2 and/or for initial e1 > 0 this limit and emax Eq. (A40)) and constant total angular momentum curves (dashed are different and the full solution of equations (A36),(A37) and 0 curves, Eq. (A39)). The initial conditions considered here are e1 = 0, (A38) is required. In fact for each initial set of e1 > 0 and itot, 0 0 g1 , e2 = 0 and L1/L2 = 0.07, appropriate for the Algol system (see there is a specific L1/L2 that will produce an angular momentum Section 5.4). We consider four different initial inclinations and their ◦ ◦ ◦ curve that crosses 90 . Thus, for initial g1 > 90 the mutual incli- symmetric 90 counterparts, from bottom to top 10, 30, 60 and 80 de- nation can oscillate from value below 90◦ to above. This happens grees. We also show an example (highlighted curve) for the system because the inclination of the outer orbit i2 changes considerably, which is a result of integration of the quadrupole-level approximation equations. while the inner orbit retains its prograde or retrograde orientation.

APPENDIX B: THE FULL OCTUPOLE-ORDER EQUATIONS OF MOTION a slight asymmetry between the prograde and retrograde orbits due to the L1/L2 factor (which is not the case for the test parti- We define: cle case, see Lithwick and Naoz 2011; Katz et al. 2011). Similar 4 9 9 6 15 k (m1 + m2) m3(m1 − m2) L1 analysis for the Algol system was done in Söderhjelm (2006, Fig- C3 = − 4 5 3 5 . (B1) 16 4 (m1 + m2 + m3) (m1m2) L G ure 1). We also write equations (A37) and (A38) using the initial 2 2 conditions. Equation (A37) can be simplified to Note that this definition differs in sign sign from Ford et al. (2000a), and is consistent with Blaes et al. (2002); Ford et al. 2 2 2 2 (2004). For m1 = m2 this factor is zero. We also define: (1 − e1) cos itot = cos i0 − 2e1 , (A40) 2 5 2 A =4+3e1 − B sin itot , (B2) depicted in Figure A1 (solid curves, for different i0). As can be 2 seen from the Figure, this equation gives the minimum eccentric- where ity, which is the crossing point with equation (A39). For these 2 2 0 B =2+5e1 − 7e1 cos(2g1) , (B3) choice of initial conditions the minimum eccentricity is e1 = 0. Equation (A38) becomes and

2 2 2 2 cos φ = − cos g1 cos g2 − cos itot sin g1 sin g2 . (B4) (1+4e1) cos itot = cos i0 + 3e1 , (A41) As mentioned in Section 4 the evolution equations for ◦ which is depicted in Figure A1 (long dashed curves, for i0 = 80 e2,g2,g1 and e1 can be found correctly from a Hamiltonian that ◦ and 100 ). We now use this equation and equation (A39) to ﬁnd has had h1 and h2 eliminated by the relation h1 − h2 = π; the

c 0000 RAS, MNRAS 000, 000–000 20 Naoz et al. partial derivatives with respect to the other coordinates and mo- The evolution of the eccentricities is: menta are not aﬀected by the substitution. The time evolution 2 1 − e1 2 of H1 and H2 (and thus i1 and i2) can be derived from the total e˙1 = C2 [30e1 sin itot sin(2g1)] (B10) G1 angular momentum conservation. Thus it is useful to write the 2 1 − e1 2 2 much simpler the doubly averaged Hamiltonian after eliminating + C3e2 [35 cos φ sin itote1 sin(2g1) the nodes: G1 2 2 2 2 − 10 cos itot sin itot cos g1 sin g2(1 − e1) H(∆h → π) = C2{ 2 + 3e1 3 cos itot − 1 (B5) − A(sin g1 cos g2 − cos itot cos g1 sin g2)] , 2 2 + 15e1sin itotcos(2 g1)} and + C3e1e2{A cos φ 2 2 2 1 − e2 2 2 + 10 cos itot sin itot(1 − e1) sin g1 sin g2} . e˙2 = −C3e1 [10 cos (itot) sin (itot) (1 − e1) sin g1 cos g2 G2 The time evolution of the argument of periapse for the inner + A(cos g1 sin g2 − cos(itot) sin g1 cos g2)] . (B11) and outer orbits are given by: We also write the angular momenta derivatives as a function 1 2 g˙1 = 6C2 [4 cos itot + (5 cos(2g1) − 1) (B6) of time; for the inner orbit G1 2 2 G˙ 1 = −C230e1 sin(2g1) sin (itot)+ C3e1e2( (B12) 2 2 cos itot 2 × (1 − e1 − cos itot)] + [2 + e1(3 − 5 cos(2g1))] 2 2 G2 − 35e1 sin (itot) sin(2g1) cos φ + A[sin g1 cos g2 1 cos itot − cos(itot) cos g1 sin g2] − C3e2 e1 + 2 2 G2 G1 + 10 cos(itot) sin (itot)[1 − e1] cos g1 sin g2) , 2 2 × [sin g1 sin g2(10(3 cos itot − 1)(1 − e1)+ A) and for the outer orbit (where the quadrupole term is zero) 2 1 − e1 − 5B cos itot cos φ] − × [sin g1 sin g2 G˙ 2 = C3e1e2[A{cos g1 sin g2 − cos(itot) sin g1 cos g2} e1G1 2 2 2 2 + 10 cos(itot) sin (itot)[1 − e1] sin g1 cos g2] . (B13) × 10 cos itot sin itot(1 − 3e1)

2 Also, + cos φ(3A − 10 cos itot + 2)] , G1 G2 H˙ 1 = G˙ 1 − G˙ 2 , (B14) and Gtot Gtot

2 cos itot 2 where using the law of sines we write: g˙2 = 3C2 [2 + e1(3 − 5 cos(2g1))] (B7) G1 sin i2 sin i1 H˙ 1 = G˙ 1 − G˙ 2 . (B15) 1 2 2 2 sin itot sin itot + [4 + 6e1 + (5 cos itot − 3)(2 + e1[3 − 5 cos(2g1)]) G2 The inclinations evolve according to 2 4e2 + 1 2 2 + C3e1 sin g1 sin g2 10 cos itot sin itot(1 − e1) H˙ 1 G˙ 1 e2G2 (cos˙ i1)= − cos i1 , (B16) G1 G1 1 cos itot 2 2 − e2 + [A + 10(3 cos itot − 1)(1 − e1)] and G1 G2 2 ˙ ˙ 1 cos itot 4e2 + 1 ˙ H2 G2 + cos φ 5B cos itote2 + + A (cos i2)= − cos i2 . (B17) G1 G2 e2G2 G2 G2 The time evolution of the longitude of ascending nodes is given Our equations are equivalent to those of Ford et al. (2000a), but by: we give the evolution equations for H1 and H2 (and i1 and i2).

3C2 2 2 h˙ 1 = − 2 + 3e1 − 5e1 cos (2g1) sin (2itot) (B8) G1 sin i1 − C3e1e2[5B cos itot cos φ APPENDIX C: ELIMINATION OF THE NODES 2 AND THE PROBLEM IN PREVIOUS − A sin g1 sin g2 + 10(1 − 3 cos itot) QUADRUPOLE-LEVEL TREATMENTS 2 sin itot × (1 − e1) sin g1 sin g2] , G1 sin i1 Since the total angular momentum is conserved, the ascending nodes relative to the invariable plane follow a simple relation, where in the last part we have used again the law of sines for h1(t) = h2(t) − π. If one inserts this relation into the Hamilto- which sin i1 = G2 sin itot/Gtot. The evolution of the longitude of nian, which only depends on h1 − h2, the resulting “simpliﬁed” ascending nodes for the outer orbit can be easily obtained using: Hamiltonian is independent of h1 and h2. One might be tempted h˙ 2 = h˙ 1 . (B9) to conclude that the conjugate momenta H1 and H2 are constants

c 0000 RAS, MNRAS 000, 000–000 Secular Dynamics in Hierarchical Three-Body Systems 21

of the motion. However, that conclusion is false. This incorrect argument has been made by a number of authors9. In general, using dynamical information about the system— in this case that angular momentum is conserved, implying that G1 + G2 = Gtot at all times and therefore h1 − h2 = π—to simplify the Hamiltonian is not correct. The derivation of Hamil- ton’s equations relies on the possibility of making arbitrary variations of the system’s trajectory, and such simplifications restrict the allowed variations to those which respect the dynamical con- straints. Once Hamilton’s equations are employed to derive equations of motion for the system, however, dynamical information can be employed to simplify these equations. In our particular case, equations of motion for components of the system that do not involve partial derivatives with respect to h1 or h2 will not be affected by the node-elimination substitution. For this reason, it is correct to derive equations of motion for all components except for H1 and H2 from the node- eliminated Hamiltonian; expressions for H˙ 1 and H˙ 2 can then be derived from conservation of angular momentum. This approach has been employed in at least one computer code for octupole evolution, though the discussion in the corresponding paper incorrectly eliminates the nodes in the Hamiltonian (Ford et al. 2000a). In some later studies, (Sidlichovsky 1983; Innanen et al. 1997; Kiseleva et al. 1998; Eggleton et al. 1998; Mikkola and Tanikawa 1998; Kinoshita and Nakai 1999; Eggleton and Kiseleva-Eggleton 2001; Wu and Murray 2003; Valtonen and Karttunen 2006; Fabrycky and Tremaine 2007; Wu et al. 2007; Zdziarski et al. 2007; Perets and Fabrycky 2009), the assumption that H1 = const (i.e. the TPQ approximation) was built into the calculations of quadrupole-level secular evolution for various astrophysical systems, even when the condition G2 ≫ G1 was not satisfied. Moreover many previous studies simply set i2 = 0. This is equivalent to the TPQ approximation; for non-test particles, given the mutual inclination i, the inner and outer inclinations i1 and i2 are set by the conservation of total angular momentum [see equations (9) and (10)].

9 For example, Kozai (1962, p. 592) incorrectly argues that “As the Hamiltonian F depends on h and h′ as a combination h − h′, the variables h and h′ can be eliminated from F by the relation (5). Therefore, H and H′ are constant.”

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