Shrinking binary and planetary orbits by Kozai cycles with tidal friction
aa r X i v : . [ a s t r o - ph ] M a y Submitted to ApJ
Preprint typeset using L A TEX style emulateapj v. 08/22/09
SHRINKING BINARY AND PLANETARY ORBITS BY KOZAI CYCLES WITH TIDAL FRICTION
Daniel Fabrycky and Scott Tremaine Submitted to ApJ
ABSTRACTAt least two arguments suggest that the orbits of a large fraction of binary stars and extrasolarplanets shrank by 1–2 orders of magnitude after formation: (i) the physical radius of a star shrinksby a large factor from birth to the main sequence, yet many main-sequence stars have companionsorbiting only a few stellar radii away, and (ii) in current theories of planet formation, the region within ∼ . ∼ ∼ d), consistent with observations indicating thatmost or all short-period binaries have distant companions (tertiaries). We also make two new testablepredictions: (1) For periods between 3 and 10 d, the distribution of the mutual inclination betweenthe inner binary and the tertiary orbit should peak strongly near 40 ◦ and 140 ◦ . (2) Extrasolar planetswhose host stars have a distant binary companion may also undergo this process, in which case theorbit of the resulting hot Jupiter will typically be misaligned with the equator of its host star. Subject headings: binaries: close — celestial mechanics — stars: planetary systems — methods:statistical INTRODUCTION
Close binaries with tertiaries
Close binary star systems (separation comparable tothe stellar radii) are often accompanied by a third star.Such triple-star systems may be stable for long times ifthe system is hierarchical, that is, if the system consistsof an “inner” binary (masses m and m ) in a nearly Ke-plerian orbit with semi-major axis a in , and an “outer”binary in which m orbits the center of mass of the innerbinary, with semi-major axis a out ≫ a in . If the forma-tion of the inner binary is independent of the formationof the outer binary, then the probability that a close bi-nary has a distant companion should be the same as theprobability that an individual star has a companion insuch an orbit. It is well known that the latter probabil-ity is substantial, about 2 / not independent: the probability of having a third compo-nent turns out to be a function of the period of the in-ner binary. Tokovinin et al. (2006) found that 96% of asample of spectroscopic binaries with periods less than3 d had a tertiary component, compared to only 34%of binaries with periods greater than 12 d. This sam-ple was not carefully selected to control biases, howeversuch biases are unlikely to change the basic result, sincethe authors observe that all of the five binaries with Electronic address: [email protected] Princeton University Observatory, Princeton, NJ 08544 Institute for Advanced Study, Princeton, NJ 08540 period less than 10 d in the volume-limited sample ofDuquennoy & Mayor (1991) have at least one additionalcompanion.A closely related observation is that the period dis-tribution of inner binaries in triple systems is quitedifferent from the period distribution of isolated bina-ries. Tokovinin & Smekov (2002) found a significantpeak at about 3 d in the logarithmic period distribu-tion of inner binaries of triple systems, a peak that isnot present in isolated binaries. Because of this fea-ture, the period distributions for binaries with and with-out a third component are different with a significanceof 0.999 (Tokovinin et al. 2006). The period distribu-tion from time domain surveys of eclipsing binaries alsopeaks at a few days (Devor 2005; Paczy´nski et al. 2006;Derekas et al. 2007), but it remains unclear whether thismay be attributed to a selection effect: binaries withlarge orbital periods have a lower probability of eclips-ing and fewer eclipses per unit time which diminishes thesignal (Gaudi et al. 2005).These observational results are surprising. In partic-ular, the median semi-major axis of the outer binary insystems with inner binary period < a in . .
07 AU)is a out ∼
70 AU (Tokovinin et al. 2006), similar tothe median for all binaries in the Duquennoy & Mayor(1991) sample. Why are the processes of star forma-tion correlated over a range of three orders of magnitudein scale? In this paper we explore the possibility thatin some circumstances the distant companion enhancestidal interactions in the inner binary, causing its periodto shrink to the currently observed value.
Kozai cycles
The study of the long-term behavior of three pointmasses interacting only through gravity has engagedphysicists and mathematicians since the time of New- Fabrycky and Tremaineton. In most cases, long-term stability requires that thesystem is hierarchical ( a out ≫ a in ). An additional re-quirement for a stable hierarchical triple system is thatthe eccentricity e out of the outer binary cannot be toolarge, so that m cannot make close approaches to m or m . An equivalent statement is that the gravitationalperturbations from m on the inner binary must alwaysbe weak. However, even weak perturbations from theouter body can have important long-term effects on theinner binary. The simplest of these is precession of the or-bital plane, which occurs if the orbital planes of the innerand outer binary are not aligned. If the inner and outerbinary orbits are circular, this precession is analogous tothe precession of two rigid rings with the same mass, ra-dius, and angular momentum as the binary orbits: boththe mutual inclination and the scalar angular momenta ofthe rings remain fixed, while the two angular-momentumvectors precess around the direction defined by the totalangular-momentum vector of the triple system.An unexpected aspect of this behavior was discoveredby Kozai (1962). Suppose the inner binary orbit is ini-tially circular, with the initial mutual inclination betweeninner and outer binaries equal to i initial . Kozai foundthat there is a critical angle i c such that if i initial is be-tween i c and 180 ◦ − i c , then the orbit of the inner binarycannot remain circular as it precesses: both the eccen-tricity of the inner binary e in and the mutual inclination i execute periodic oscillations known as Kozai cycles. Theamplitude of the eccentricity and inclination oscillationsis independent of the strength of the perturbation fromthe outer body, which depends on m , a out , and e out , butthe oscillation amplitude does depend on i initial : for ini-tial circular orbits with i initial = i c or 180 ◦ − i c , the max-imum eccentricity is 0, but if i initial = 90 ◦ the maximumeccentricity is unity; i.e., the two inner bodies collide.Kozai cycles can be investigated analytically by av-eraging over the orbital phases of the inner and outerbinaries (Kozai 1962; Ford et al. 2000); this averaging,usually called the secular approximation, is justified be-cause the precession time is generally much longer thanthe orbital time of either binary. In the averaged prob-lem the semi-major axes of the inner and outer binary areboth conserved. The analysis is simplest in the limitingcase when a out ≫ a in (so that the perturbing potential ofthe outer body can be written in the quadrupole approx-imation) and the angular momentum of the outer binaryis much greater than that of the inner binary (so that theorientation of the outer binary is a constant of the mo-tion). With these approximations, the following resultshold. (i) The averaged quadrupole potential from theouter binary is axisymmetric relative to its orbital plane.(ii) The averaged problem can be described by a Hamil-tonian with one degree of freedom. (iii) The eccentricityof the outer binary is constant. (iv) The critical inclina-tion is i c = cos − p / . ◦ . (v) If the inner orbitis initially circular the maximum eccentricity achievedin a Kozai cycle is e in,max = [1 − (5 /
3) cos i initial ] / .(vi) Depending on the initial conditions, the argument ofpericenter ω in (the angle measured in the orbital planebetween the pericenter of the inner binary and the or-bital plane of the outer binary) can either librate (os-cillate around 90 ◦ or 270 ◦ ) or circulate. The systemmay remain at a fixed point with ω in = 90 ◦ or 270 ◦ and e in = [1 − (5 /
3) cos i fix ] / if i c < i fix < ◦ − i c . (vii)The only property of the Kozai oscillation that dependson the masses of the three bodies, their semi-major axes,or the eccentricity of the outer binary is the period of theoscillation, which is of order the timescale (Kiseleva et al.1998): τ = 2 P out πP in m + m + m m (1 − e out ) / ; (1)small-amplitude libration about the fixed point takesplace with a period P lib = τ π √ − (5 /
3) cos i fix ) / sin i fix . (2)(viii) Octupole and higher-order terms in the perturb-ing potential introduce a narrow chaotic zone aroundthe separatrix between circulating and librating solu-tions as determined by the quadrupole approximation(Holman et al. 1997).Consider a sequence of triple systems in which thesemi-major axis a out of the outer binary becomes largerand larger, while its mass, inclination, and eccentricityremain the same. The maximum eccentricity of the in-ner binary in the Kozai cycle will remain fixed, but theperiod of the Kozai cycle will grow as a out . This behav-ior will continue so long as the perturbation from theouter body is the dominant cause of apsidal precessionin the inner binary orbit. Thus, weak perturbations fromdistant third bodies can induce large eccentricity and in-clination oscillations. However, small additional sourcesof apsidal precession in the inner binary—general relativ-ity, tides, the quadrupole moments of the two members ofthe inner binary, planetary companions, etc.—can com-pletely suppress Kozai oscillations caused by a distantthird body if they dominate the apsidal precession. Tides
The dissipative forces due to tides on the stars of theinner binary are only significant if the two stars are sepa-rated by less than a few stellar radii. Thus the tidal fric-tion on an inner binary with a semi-major axis of (say)0 .
25 AU, corresponding to an orbital period of 32 d if m = m = 1 M ⊙ , is normally negligible. However, if theamplitude of the eccentricity oscillation during a Kozaicycle is sufficiently large, the pericenter distance of theinner binary may become sufficiently small at some phaseof the cycle that tidal friction drains energy from the or-bit, reducing the semi-major axis and thereby enhancingthe friction, until the inner binary settles into a circularorbit with a semi-major axis of only a few stellar radii.Following Tokovinin et al. (2006), we refer to this processas Kozai cycles with tidal friction (KCTF). Previous work
Harrington (1968) first suggested that KCTF mightbe an important evolutionary mechanism for triplestars. Mazeh & Shaham (1979) showed how the long-period perturbations of a third body could reducethe inner binary’s separation on the tidal dissipationtimescale. Kiseleva et al. (1998) used KCTF to con-strain the strength of tidal dissipation in stars and fo-cused on the possibility that the inner binary of the Al-ozai cycles with tidal friction 3gol system might have shrunk significantly by this mech-anism. Eggleton & Kiseleva-Eggleton (2001) presentedthe differential equations that we use here to modelKCTF, which parametrize the extra forces (tidal fric-tion, quadrupole from a distant companion, etc.) actingon a binary orbit.Kozai cycles have also been studied in the planetarycontext. Mazeh et al. (1997), Innanen et al. (1997), andHolman et al. (1997) all suggested that the large ec-centricities observed in many extrasolar planet orbitscould be explained by Kozai cycles if the host starwere a member of a binary system; however, this hy-pothesis leads to two predictions that are not verified(Tremaine & Zakamska 2004): (i) high-eccentricity plan-ets should mostly be found in binary systems; (ii) multi-planet systems should have low eccentricities, since theirmutual apsidal precession suppresses the Kozai cycle.Also, Takeda & Rasio (2005) argued that even if ev-ery extrasolar planet host has an undetected compan-ion, Kozai cycles alone cannot explain the distributionof observed eccentricities. (They found that too manyplanets remained on nearly circular orbits, but that re-sult is sensitive to the initial eccentricity assumed sincethe time spent near the unstable fixed points at e in = 0, ω in = ± ◦ , ± ◦ dominates the period of the Kozai cy-cle, and this time depends strongly on the initial eccen-tricity; see Innanen et al. 1997.) Kiseleva et al. (1998)speculated that KCTF might explain the presence ofmassive planets on orbits close to their parent stars(“hot Jupiters”), a possibility that we re-examine in thispaper. Wu & Murray (2003) elaborated the specula-tion of Kiseleva et al. (1998) for the orbit of the planetHD 80606b, which is unusual because of its very largeeccentricity ( e = 0 .
93) and small pericenter distance( a (1 − e ) = 0 .
033 AU).Blaes et al. (2002) have examined a process similarto KCTF for triple black-hole systems that might befound in the centers of galaxies; here the dissipativeforce is gravitational radiation rather than tidal frictionbut much of the formalism is the same. A major dif-ference is that tidal friction vanishes in a binary witha circular orbit and synchronously rotating stars, whilegravitational radiation does not. Therefore the end-stateof a black-hole triple subject to the analog of KCTF isa merger, leaving a binary black hole (which may be-come unbound by the gravitational radiation recoil ofthe merger). In the present application, however, gravi-tational radiation is negligible. Therefore the stellar andplanetary cases offer opportunities to compare the orbitsof the observed systems to the distributions predicted bythe theory, which this paper quantifies.
The plan of this paper
In this paper we shall model KCTF using (i) the secularapproximation for orbital evolution; (ii) the quadrupoleapproximation for the tidal field from the outer body;(iii) a simple model for apsidal precession that includesthe dominant general relativistic precession term and thequadrupole distortion of the stars of the inner binary dueto tides and rotation; (iv) a simple model for orbital de-cay due to tidal friction; (v) the assumption that theouter binary contains most of the angular momentum inthe system. Our main goal is to characterize the statis-tical properties of the binary systems that result from KCTF. In § § §
4. We numerically integrate a large number oftriple systems in §
5, compare the final conditions to anempirical period distribution, and give a prediction forthe mutual inclination distribution. In § §
7. We discuss applications and possible extensions ofour analysis in § § EQUATIONS OF MOTION
The differential equations that govern the inner bi-nary’s orbital parameters and the two stars’ spin param-eters were presented by Eggleton & Kiseleva-Eggleton(2001) and are recalled here. The equations take intoaccount stellar distortion due to tides and rotation, tidaldissipation based on the theory of Eggleton et al. (1998),relativistic precession, and the secular perturbations ofa third body (averaged over the inner and outer Ke-plerian orbits). The orientation of the inner orbit isspecified by its Laplace-Runge-Lenz vector, e in , whosemagnitude is the inner eccentricity e in and whose di-rection is towards the pericenter of the orbit of m about m . Also needed is the specific angular momen-tum vector h in of the inner orbit, whose magnitude is[ G ( m + m ) a in (1 − e in )] / , where G is the gravita-tional constant. The vector ˆq in = ˆh in × ˆe in completesthe right-hand triad of unit vectors ( ˆe in , ˆq in , ˆh in ). Eachof the stars of the inner binary also has a spin vector Ω j , j = 1 , e in d e in dt = ( Z + Z + Z GR ) ˆq in − ( Y + Y ) ˆh in − ( V + V ) ˆe in − (1 − e in )[5 S eq ˆe in − (4 S ee − S qq ) ˆq in + S qh ˆh in ] , (3)1 h in d h in dt = ( Y + Y ) ˆe in − ( X + X ) ˆq in − ( W + W ) ˆh in + (1 − e in ) S qh ˆe in − (4 e in + 1) S eh ˆq in + 5 e in S eq ˆh in , (4) I d Ω dt = µh in ( − Y ˆe in + X ˆq in + W ˆh in ) , (5) I d Ω dt = µh in ( − Y ˆe in + X ˆq in + W ˆh in ) . (6)Subscripts 1 and 2 refer to the masses m and m . Thegravitational influence of the third body is described bythe tensor S . We specify the orientation of the orbit ofthe third body using the triad ( ˆe out , ˆq out , ˆh out ), definedby analogy with the inner binary’s coordinate system,treating the inner binary as a point mass m + m at itscenter of mass. We have: Fabrycky and Tremaine S mn = C ( δ mn − h out,m ˆ h out,n ) (7) C = 2 π τ (1 − e in ) − / (8)Here the directions m, n ∈ (ˆ e in , ˆ q in , ˆ h in ) are along thebasis vectors of the inner orbit, and ˆ h out,m and ˆ h out,n aredirection cosines of the outer orbit’s angular momentumalong the inner orbit’s coordinate directions. P in and P out are the periods of the inner and outer orbits, respec-tively. V and W are dissipation rates for e and h , respec-tively. The vector ( X, Y, Z ) in the ( ˆe in , ˆq in , ˆh in ) frame isthe angular velocity of that frame relative to the inertialframe. Z GR is the first post-Newtonian effect of relativ-ity which causes the pericenter to precess. The stars ofthe inner binary have moment of inertia I and I , and µ = m m / ( m + m ) is the reduced mass of the innerbinary. The functional forms of the V , W , X , Y , and Z terms were modeled in Eggleton & Kiseleva-Eggleton(2001) and for convenience are repeated in the appendix.To check the code we use to integrate these equations,we have replicated the results of Wu & Murray (2003),who studied KCTF for the planet HD80606b, whose hoststar has a binary companion with a out ∼ AU. Werecalculate their Figure 1, using their specified param-eters and initial conditions (Fig. 1). We started boththe planet’s and the star’s obliquity (the angle betweenˆ h in and Ω or Ω ) at 0 ◦ . Prominent eccentricity oscil-lations are seen in panel (a). The energy in the planet’sspin was transferred to the orbit, increasing the semi-major axis for the first 0 . . ψ is substantial. After a in equilibrates,a large amount of misalignment will generally remain forthe main-sequence lifetime of the star, although slightmovement towards alignment occurs because of dissipa-tive tides being raised on the star by the planet. Thetidal field of the planet applies a torque to the rotationalbulge of its host star, which causes the spin axis to pre-cess; compared to the fixed reference frame of the thirdbody’s orbit, the host star’s spin axis oscillates with largeamplitude (panel f). To conserve angular momentum,the planet’s orbit likewise precesses, but on a consid-erably smaller scale—it is the few degree wiggle in theinclination after migration (after 2 . § § ANALYTIC UNDERSTANDING OF KCTF
In this section we seek an approximate analytical the-ory of KCTF. To lowest order in a in /a out , the ter-tiary component presents a quadrupole tide to the in- ner binary. As a in shrinks by KCTF, this approxima-tion will become better and better. The instantaneousquadrupole-order Hamiltonian is (Ford et al. 2000): F = − Gm m a in − G ( m + m ) m a out + F q (9) F q = − Gm m m m + m r in r out P (cos Φ) , (10)where r in is the vector from m to m , r out is the vectorfrom the inner binary center of mass to m , Φ is the anglebetween r in and r out , and P ( x ) = x − .This quadrupole-order potential may be integratedover the unperturbed motion in both orbits to removeshort period terms, which depend on the two orbits’mean anomalies. However, as in Innanen et al. (1997),we shall work in the approximation that the outer or-bit contains essentially all the angular momentum andhence is fixed, therefore the mutual inclination i is thesame as the inclination relative to the total angular mo-mentum ( i in = i , i out = 0). The averaged Hamiltonianis (Innanen et al. 1997; Ford et al. 2000): hFi = − Gm m a in − G ( m + m ) m a out + hF q i (11) hF q i = − Gm m m m + m a in a out (1 − e out ) / × (cid:16) e in − (3 + 12 e in − e in cos ω in ) sin i (cid:17) , (12)where the orbital elements are referenced to the plane ofthe outer binary. The canonical Delaunay variables forthe inner binary are the mean anomaly l in , argument ofpericenter ω in , and longitude of the ascending node Ω in ,along with their respective canonical momenta L in = m m p Ga in / ( m + m ), G in = L in p − e in = µh in ,and H in = G in cos i . The canonical equations of motionfor the inner orbit are:˙ l in = ∂ hFi ∂L in , ˙ L in = − ∂ hFi ∂l in , (13)˙ ω in = ∂ hFi ∂G in , ˙ G in = − ∂ hFi ∂ω in , (14)˙Ω in = ∂ hFi ∂H in , ˙ H in = − ∂ hFi ∂ Ω in , (15)with corresponding equations for the evolution of the co-ordinates and momenta of the outer variables.The averaging procedure removes the Hamiltonian’sdependence on the mean anomalies l in and l out , so L in and L out and thus the semi-major axes of the inner andouter orbits are conserved. The average Hamiltonian hFi is conserved, because the Hamiltonian is independent oftime; moreover hF q i depends only on hFi and the con-served semi-major axes (eq. [11]), so hF q i is also con-served. Another conserved quantity is H in , because theHamiltonian is independent of its canonical conjugate,Ω in . A final conserved quantity is the eccentricity of theouter binary, e out , because the Hamiltonian is indepen-dent of ω out .ozai cycles with tidal friction 5 Fig. 1.—
The evolution of a planet initially in an orbit with a in = 5 AU, i = 85 . ◦ , e in = 0 . ω in = 45 ◦ , as a hypothetical progenitorto HD 80606b. The spins of both the planet and its host star were initialized with zero obliquity. The stellar companion was assumed tohave m = 1 . m ⊙ , a out = 1000 AU, and e out = 0 .
5. The values of structural parameters were the same as those used by Wu & Murray(2003), and the viscous times were t V,star = 50 yr and t V,planet = 0 .
001 yr (see appendix). Energy dissipation is dominated by the planet.The diamonds mark the current position of HD 80606 along this possible evolution.
We may write dimensionless versions of the conservedquantities as: F ′ = − − e in +(3 + 12 e in − e in cos ω in ) sin i (16) H ′ = (1 − e in ) / cos i. (17)Kiseleva et al. (1998) have given a different conservedquantity which is simply a combination of the above:(5 − H ′ + F ′ ) / e in,initial , ω in,initial , i initial ). It is then pos-sible to compute analytically the values of e in and i atany value of ω in accessible to the system. Let us restrictour discussion to i initial > ◦ (for retrograde systems,identical behavior of the inclination, mirrored across 90 ◦ ,results). The system attains a maximum e in and mini- mum i when ω in = 90 ◦ or 270 ◦ . Therefore, we have: e in,max = (cid:16) { [(10 + 12 H ′ − F ′ ) − H ′ ] / +8 − H ′ + F ′ } / (cid:17) / (18) i min = cos − [ H ′ (1 − e in,max ) − / ] . (19)If F ′ is initialized with e in = 0, the value of e in,max given by equation (18) matches the value quoted in theintroduction.Figure 2a is a contour plot of the minimum mutualinclination attained as a function of initial eccentricityand argument of pericenter. For an initial eccentricity e in,initial < .
2, the minimum inclination is within 2 ◦ of the critical inclination, i c = 39 . ◦ . Integrating overa uniform initial distribution of arguments of pericen-ter and various initial eccentricity distributions gives the Fabrycky and Tremaineprobability distributions of minimum inclination shownin Figure 2b. These results are only weakly dependenton the initial inclination.We now argue that the minimum inclinations shown inFigure 2a are very nearly equal to the final inclinationsproduced by KCTF. If we approximate the effect of tidaldissipation as acting only when the inclination is at itsminimum (i.e., eccentricity maximum), then i min is con-served between Kozai cycles (Fig. 1c) even though theconstants F ′ and H ′ are not conserved in the presenceof tidal friction. Eventually, all eccentricity is damped(after 4 Gyr in Fig. 1a) and the system finishes with aninclination nearly equal to its minimum inclination fromthe first cycle (Fig. 1c). The distributions of minimuminclination shown in Figure 2b will correspond to the fi-nal inclinations after KCTF if the above approximationsare correct. Plotted for comparison is the inclinationdistribution of an isotropic distribution of triples, i.e.,triples with inner and outer angular momentum direc-tions uncorrelated. The distributions clearly distinguisha population of triples whose inner binaries have shrunkby KCTF from a population with its primordial inclina-tions, if these are uncorrelated.This calculation will only be a faithful representation ofthe final inclination distribution if pericenter precessiondue to other causes is negligible compared to the pre-cession caused by the tertiary component. Under theseconditions, the final distribution of mutual inclinationshows strong peaks close to the critical angle for Kozaicycles: 39 . ◦ or 140 . ◦ , as seen in Figure 2b. For a morerealistic calculation considering a population of triplesand including the extra precession forces, see § Kozai cycles in the presence of extra forces
Because Kozai cycles are driven by the interplay be-tween the weak tidal forces from the outer binary andthe shape of the orbit of the inner binary, they canbe easily suppressed by other weak effects leading topericenter precession in the inner binary. Mathemati-cally, the rate of change of eccentricity is proportional to − S eq = 3 C ˆ h out,e ˆ h out,q ∝ sin 2 ω in (according to eq. [3]and eq. [7]). If ω in is changing too fast, S eq will averageto zero before e in has time to grow. We consider fourcauses of extra pericenter precession—relativity, tides,stellar rotational distortion, and extra bodies. Here wecharacterize the first three effects with additional Hamil-tonian terms and calculate the associated apsidal motion˙ ω in . For extra Hamiltonian terms hF extra i , equation (14)gives:˙ ω extra = ∂ hF extra i ∂G in = − (1 − e in ) / e in L in ∂ hF extra i ∂e in . (20)First, let us consider relativity. As inEggleton & Kiseleva-Eggleton (2001), we only con-sider pericenter precession, which is the largest effect.Take an extra Hamiltonian term of hF GR i ( e in ) = − G m m ( m + m ) a in c − e in ) / , (21)which depends on both L in and G in . The former de-pendence gives an additional contribution to the mean Fig. 2.— (a) The minimum inclination during a Kozai cycle,starting with i initial = 86 ◦ at various eccentricities e in,initial andarguments of pericenter ω in,initial for the inner orbit (solid lines).These results depend very weakly on the initial inclination; for i initial = 70 ◦ , the contours for i min = 25 ◦ , 38 ◦ , 41 ◦ , and 65 ◦ (dot-ted lines) are only slightly different. (b) The inclination minimumdistribution (probability density function), assuming ω in,initial isuniformly distributed in angle. These distributions result fromthree different assumptions for the initial eccentricity distribution:circular orbits, uniform distribution in eccentricity e in,initial , anduniform distribution in e in,initial (constant phase-space density onthe energy surface). They were computed with i initial = 86 ◦ , butall initial inclinations near 90 ◦ (i.e., all systems that evolve sub-stantially by KCTF) produce similar distributions. If KCTF sealsin this inclination minimum (see text), the distribution of tripleswill show spikes at the critical angles i c ≃ ◦ and 180 ◦ − i c ≃ ◦ and very few systems near 90 ◦ . An isotropic distribution, with nocorrelation between the directions of inner and outer orbital an-gular momenta, is plotted for comparison. See also Figure 7b forthe inclination distribution that results from integrating the fullequations of motion for a population of triples. motion, which is incorrect but does not affect the secularresults. The latter gives˙ ω GR = 3 G / ( m + m ) / a / in c (1 − e in ) , (22)which is the standard expression for the rate of pericenterprecession due to relativistic effects.Next let us consider how the non-dissipative tidal bulgecontributes to the apsidal motion. At any moment, con-struct a spherical polar coordinate system centered on m with radius r and a polar angle θ ′ measured fromthe vector ˆ r in . The tidal potential of m to lowest orderin r /r in is φ ,t = − Gm P (cos θ ′ ) r r − in . (23)The surface of m is an equipotential. The correspondingdistortion produces an external potential of φ ,ep = − Gm P (cos θ ′ ) r − r − in R k , (24)where R j is the radius and k j is the classical apsidalozai cycles with tidal friction 7motion constant (Russell 1928) of the j th body. Typicalvalues of k j are 0 .
014 for stars and 0 .
25 for gas giantplanets. Back at m (for r = r in , θ ′ = π ), the extraforce per unit mass is − ∂φ ,ep ∂r (cid:12)(cid:12)(cid:12) r = r in = − Gm r − in R k , (25)which can be integrated to find the potential associatedwith the tidal distortion of m by m : φ , = − Gm r − in R k . (26)Notice this is a factor of 2 smaller than equation (24)evaluated at the location of m because the derivative of φ ,ep is taken with respect to the spatial coordinate r ,but φ , is assembled by integrating equation (25) withrespect to r in (see, e.g., Sterne 1939 and the Appendixof Eggleton & Kiseleva-Eggleton 2001). By analogy withthe theory of image charges in electrostatics, the effectivepotential is half the physical potential (P. Eggleton, pri-vate communication). From this potential we may formthe instantaneous Hamiltonian F T ide, = m φ , . Afteraveraging over one orbit of the inner binary and account-ing for both stars, we have hF T ide i ( e in ) = − G a in e in + 3 e in (1 − e in ) / × h m k R + m k R i . (27)After converting to canonical variables, the equations ofmotion yield:˙ ω T ide = 15( G ( m + m )) / a / in e in + e in (1 − e in ) × h m m k R + m m k R i . (28)This expression is always positive, so tidal bulges alwaystend to promote pericenter precession and therefore sup-press Kozai oscillations.An analogous procedure gives the extra piece of theHamiltonian arising from the rotational bulges of thestars of the inner binary. The instantaneous Hamilto-nian resulting from the quadrupole field of the rotation-ally oblate m is: F Rotate, = k m Ω R r in P (cos θ ) , (29)where θ is the angle measured from the spin axis of m . Averaging the result, accounting for both stars andputting Ω and Ω in component form, hF Rotate i ( e in , Ω , Ω ) = − m m a in (1 − e in ) / × X j =1 , k j R j m j (2Ω jh − Ω je − Ω jq ) . (30)To use the canonical equations of motion, each Ω j com-ponent must be converted to components in the inertialframe. After taking a derivative with respect to G in and converting back to the orbit frame, we have:˙ ω Rotate = ( m + m ) / G / a / in (1 − e in ) × X j =1 , k j R j m j h (2Ω jh − Ω je − Ω jq )+ 2Ω jh cot i (Ω je sin ω in + Ω jq cos ω in ) i . (31)Small additional bodies in the system, such as plan-ets, also affect the dynamics through their contributionto the apsidal precession rates of the stars in the in-ner binary. In binary stellar systems, planetary orbitsmay stay close to one of the stars (S-type orbits, for“satellite”) or encompass the whole binary (P-type or-bits, for “planetary”). Stable S-type or P-type orbitsmust obey certain stability criteria (Holman & Wiegert1999; Mardling & Aarseth 2001; Mudryk & Wu 2006);crudely, these require that the semi-major axis a ≪ a in for stable S-type orbits, while a ≫ a in for stable P-typeorbits. In triple stellar systems, an analogous classifica-tion scheme can be worked out. Three different types oforbits may be stable: (1) S-type about any of the stars,(2) P-type with respect to the inner binary, but S-typewith respect to the outer binary, and (3) P-type withrespect to the outer binary. In the first case, the time-average of an S-type orbit in the equatorial plane of itshost star will qualitatively act as an additional contri-bution to stellar oblateness. Although large eccentricityoscillations of the inner binary would tend to destabilizeS-type planets, the extra pericenter precession caused bythe planet may suppress those oscillations: even a tinyplanet may thereby be responsible for its own survival.In case (2) the outer binary can induce Kozai oscillationsin the planetary orbit, while in case (3) the planet caninduce Kozai oscillations in the outer binary. The con-tribution of such additional bodies to Kozai oscillationshas been studied in the context of multi-planet S-typesystems in binaries (Innanen et al. 1997; Wu & Murray2003; Malmberg et al. 2006), but the huge parameterspace of the general secular four-body problem has notyet been systematically explored. Modified eccentricity maximum
When the additional forces described in § e max as before, although theconserved Hamiltonian is now hF tot i = hF q i + hF extra i , (32)where hF extra i = hF GR i + hF T ide i + hF Rotate i . If hF Rotate i contributes significantly, one must take into ac-count that each component of the spins (Ω i , Ω i ) willbe a function of time.Let us consider the case in which hF extra i is dominatedby hF GR i and assume that the orbit is initially circularto evaluate hF tot i . Eccentricity maximum still occursat ω in = 90 ◦ or 270 ◦ , which simplifies the expressionfor hF q i at eccentricity maximum. The conservation of hF tot i and H ′ (eq. [17]) gives an implicit equation for e in,max : Fabrycky and Tremaine Fig. 3.—
The maximum eccentricity attained by systems withinitially circular orbits of varying initial inclinations, including rel-ativistic precession (eq. [33]). The curves are parameterized by therelative strength of relativistic precession to that of the tidal fieldof the third body: τ ˙ ω GR ˛˛ e in =0 , where τ is defined by equation (1)and ˙ ω GR (eq. [22]) is evaluated at e in = 0. As the timescale of rel-ativistic precession becomes as short as the timescale of precessioninduced by the companion, the eccentricity cycles are reduced inamplitude and the critical inclination—the largest i initial for whichthe orbit remains circular—grows (eq. [35]). In contrast to the sit-uation without relativistic precession, a system in which the innerand outer binary orbits are initially perpendicular ( i initial = 90 ◦ )will not reach a radial orbit but attain a more moderate eccen-tricity (see eq. [34]). No Kozai oscillations occur for any initialinclination if τ ˙ ω GR ˛˛ e in =0 ≥ cos i initial = (1 − e in,max ) − [(1 − e in,max ) − / + 1] − τ ˙ ω GR (cid:12)(cid:12) e in =0 , (33)where the strength of relativity is parametrized by theproduct of the Kozai timescale, τ (eq. [1]) and ˙ ω GR eval-uated at e in = 0 (eq. [22]). We plot this function inFigure 3. There is a maximum eccentricity that can bereached for a given amount of GR precession; beginningwith mutual inclination of 90 ◦ and negligible eccentricity: e in,max = { − [( + τ ˙ ω GR (cid:12)(cid:12) e in =0 ) / − / } / . (34)The critical inclination for eccentricity oscillations isalso increased by relativity:cos i c = 3 / − (1 / τ ˙ ω GR (cid:12)(cid:12) e in =0 . (35)In summary, some hierarchical triples, even if they beginwith perpendicular orbits, may avoid close encountersbecause relativistic precession suppresses Kozai cycles.A similar analysis has been performed by Blaes et al.(2002). TIDAL SHRINKAGE IN ISOLATED BINARIES
In this section, we select a sample of isolated binariesvia Monte Carlo methods, then we follow how their or-bits evolve by tidal dissipation. This calculation providesa control sample for the numerically integrations of thefollowing section, in which we follow the secular evolutionof triple stars by KCTF.The initial orbital distributions were taken from theobserved distributions of Duquennoy & Mayor (1991). For simplicity, m was set to M ⊙ . Next, m waschosen by selecting the mass ratio q in = m /m from a Gaussian distribution with mean 0.23 and stan-dard deviation 0.42, as found by Duquennoy & Mayor(1991) (negative results were resampled). The starswere given radii consistent with their masses for starson the main sequence: R = R ⊙ and R = R ⊙ ( m /M ⊙ ) . (Kippenhahn & Weigert 1994). Theirstructural constants were set to typical values given inEggleton & Kiseleva-Eggleton (2001): k = k = 0 . t V = t V = 5 yr (vis-cous timescales; see appendix), I = 0 . M R and I = 0 . M R (moments of inertia). The initial spin pe-riods were each 10 d, with spin angular momenta alignedwith the binary’s orbital angular momentum.The orbital period was picked from the log-normal distribution of Duquennoy & Mayor (1991) (with h log P [d] i = 4 . σ log P [d] = 2 . . P < e was cho-sen from a Rayleigh distribution ( dp ∝ e exp( − λe ) de )with h e i / = (2 λ ) − / = 0 .
33; and for
P > e was chosen from an Ambartsumian distribution ( dp =2 ede ), corresponding to a uniform distribution on theenergy surface in phase space. One might suppose thatthe observed eccentricity distribution is peaked towardssmall values for P < < P < e > .
5. However, tidal dissipation wouldonly be able to affect the orbit of a main-sequence bi-nary with P ≈
100 d if e & .
8. Therefore, the smalleccentricities of binaries with
P < × such binaries were selected, and theevolution of P , e , Ω and Ω were followed using theequations of §
2. The integrations proceed very quickly,since the spins were assumed to be aligned with theorbit—only the magnitudes of the vectors in equations(3)-(6) needed to be followed, not their orientation. Theintegrations were stopped when e ≤ − or at 10 Gyr(roughly the main sequence lifetime of a 1 M ⊙ star),whichever came first.In Figure 4 we plot the initial and final period distri-butions. The Gaussian tail at long periods is not plot-ted. Histograms corresponding to various initial periodranges are plotted in different shades. In particular, ifthe primordial period distribution were cut off below 6 d,both the initial and final histograms would consist only ofthe two darker shades. The unshaded region is probablynot physical—main-sequence binaries with such periodswould merge.A firm result is that tidal dissipation alone causes verylittle change to the period distribution of binaries. Weozai cycles with tidal friction 9 Fig. 4.—
The periods of isolated binaries before and after 10 Gyrof evolution by tidal friction. (a) Histogram of the assumed initialperiod distribution. (b) Histogram of the final period distribution,showing no dramatic change.
Fig. 5.—
The periods of the inner binaries of the simulated triplesbefore and after 10 Gyr of evolution by Kozai cycles with tidal fric-tion. (a) Histogram of the assumed initial inner period distribution.(b) Histogram of the final period distribution, showing the pro-duction of numerous close binaries with 0 . . P in,final .
10 d,many of which initially had much longer periods. turn now to KCTF. POPULATION OF TRIPLES
In this section we calculate the semi-major axis and in-clination distributions that result from KCTF. We gen-erated systems by Monte Carlo methods, then integratedtheir equations of motion ( § Initial conditions
The initial orbital distributions for both the inner andouter binaries were taken from the observed distributionsof Duquennoy & Mayor (1991). Both the orbital distri-butions of the binaries and the physical parameters ofthe stars of the inner binary are chosen as above ( § m was determined by choosing q out = m / ( m + m ) from a Gaussian distribution as for q in .This approach implies that the mass of the third star wascorrelated with the mass of the inner binary, but we donot believe that this correlation has any significant effecton our results.Two periods and eccentricities were picked as above.The smaller (larger) period was assigned to the inner(outer) orbit. The semi-major axes were computed fromthese masses and periods assuming non-interacting Kep-lerian orbits. The mutual inclination distribution of thetertiary is assumed to be isotropic with respect to the in-ner binary; thus we selected cos i to be uniform in [ − , ω in and Ω in , were selected uniformlyin [0 , π ].After these parameters were selected, we used the em-pirical stability criterion of Mardling & Aarseth (2001)to determine whether the system is hierarchical or if itwill disrupt in a small number of dynamical times. If thesemi-major axes obeyed the criterion: a out /a in > . q out ) / (1 + e out ) / (1 − e out ) / (1 − . i/ ◦ ) , (36)then we accepted the triple as stable and integrated itsaveraged equations of motion. Otherwise, we assumedit disrupted, resulting in an unbound binary and singlestar (we do not include those binaries in the followingresults). About 40% of selected triples failed to fulfillthe condition (36). A total of 7 × stable systemswere integrated and the results are presented here. Stopping conditions
In most cases we stopped the integrations at 10 Gyr,roughly the main sequence lifetime of a 1 M ⊙ star. How-ever, for some systems a straight-forward integration ofthe averaged equations of motion was prohibitively ex-pensive. In these cases we used the following procedureto deduce the final state without a costly integration.The largest such group is triples whose Kozai cycledoes not cause pericenter passages close enough for tidaldissipation to be effective. For these we integrated theequations until the first eccentricity maximum and com-puted the eccentricity damping timescale ( V + V ) − (eq. [A1]) there. If it was longer than 10 Gyr, we in-tegrated until a second eccentricity maximum, then tookthe properties at a random time in the interval betweenthe maxima, similar to the method of Takeda & Rasio(2005). These systems will oscillate for their whole mainsequence lifetimes, so choosing a random point of an os-cillation near the initial time is statistically indistinguish-able from a random time at the currently observed epoch.If the triple is strongly hierarchical initially, or if it isdriven to such a state by KCTF, then the pericenter pre-cession due to relativity and stellar distortion dominatesthat of the third body. As shown in § § ∼
100 Kozai cy-cles. For some individual systems we checked that thefinal state of this integration had parameters to within apercent of those of the final states of the original systems.These systems either reached the end of their allottedtime, which was scaled down from 10 Gyr in proportionto the scaling of the viscous time, or stopped oscillating,which allowed the neglect of the third body in integratingthe further evolution as above.
Numerical results
Figure 5 shows the relation between the initial andfinal period distributions for the inner binary. Shadedportions of the histogram show how initial periods mapto final periods. The main result is the strong peak in thedistribution of periods near P in,final ≃ .
25, and the best-fitting primordial cutoff period is 6 d.These parameters are consistent with independent esti-mates. The observations of additional components to bi-nary stars are quite incomplete, but approximately one-third of visual binary stars have an additional compo-nent (Tokovinin & Smekov 2002; Tokovinin 2004), con-sistent with our estimate. A complication is that theobserved mutual inclination distribution of triple starswith long-period inner binaries shows moderate correla-tion between the inner and outer orbits (Tokovinin 1993).Recent data suggests h i i ≈ ◦ ± ◦ (Sterzik & Tokovinin2002), and the distribution can be reasonably representedby the sum of an isotropic distribution (cos i uniform in[ − , i uniform in [0 ◦ , ◦ ]; 25% of systems). Theformer are the systems we have simulated, and the lat- Fig. 6.—
The fraction of binaries with tertiaries. The points arefrom the observational study of Tokovinin et al. (2006)—horizontalbars indicate the period range and vertical bars represent the er-ror on the fraction of tertiaries. The gray theoretical histogram isconstructed via a linear combination of the final distributions com-puted in § §
5. Two free parameters were varied to achievebest fit with the observational results: (a) the overall fraction oftriples relative to all systems (binaries plus triples)—0.25 was best-fit—and (b) the cutoff period of the primordial distribution—6 dwas best-fit, corresponding to histograms in Figures 4b and 5bincluding only the two darkest shades. ter do not evolve substantially by KCTF. Therefore, dueto this correlation, our overall tertiary fraction of 0 . ∼ /
3, inwhich case it is still consistent with the triple frequencyestimated by Tokovinin. Similarly, our best-fitting pe-riod cutoff is consistent with the size of protostars, oncethey become dynamically stable, only contracting on theKelvin-Helmholtz timescale. For example, a solar massstar at an age of 7 × yr has a radius of ∼ R ⊙ (D’Antona & Mazzitelli 1994); a binary consisting of twosuch stars in marginal contact would have a period of3 . P in,final be-tween 3 and 10 d. During Kozai cycles, the inclinationtends to move away from 90 ◦ . Tidal dissipation at max-imum eccentricity seals in these more moderate inclina-tions. The most interesting feature in the distribution of i final (Fig. 7b) is the spikes that appear near i c = 39 . ◦ and 140 . ◦ . The shaded histograms show the result bro-ken down by initial inner binary period. Triple systemsthat start with P in,initial between 3 d and 10 d, butdo not evolve by KCTF, contribute a rather isotropicdistribution (lightest shading). Therefore the spikes arefractionally much stronger if the primordial inner binaryperiod distribution is cut off at a large period. This in-clination distribution is a distinctive feature of Kozai cy-cles, as predicted by the simple model of § Fig. 7.— (a) The initial and final mutual inclinations for innerbinaries with final periods between 3 d and 10 d. Systems moveaway from 90 ◦ during the high eccentricity phase of a Kozai cy-cle, so tidal dissipation during that phase tends to lock in a moremoderate mutual inclination. (b) Histogram of final mutual in-clination for these systems. The strong spikes correspond to themost probable mutual inclinations at the high eccentricity portionof Kozai cycles (compare to Fig. 2b). These spikes are strongestfor binaries that have shrunk dramatically by Kozai cycles; e.g., ifprimordial binaries with P in,initial < ◦ increment of i are expected to inhabit thespikes relative to other configurations. and therefore would provide unambiguous observationalevidence for KCTF.In a small fraction of triples, the third star is closeenough ( . i final = 0 ◦ or i final = 180 ◦ (seeFig. 7a, but the majority of systems that undergo thisprocess have P in,final < H in is still approximately conserved (onlya small amount of angular momentum is transfered tothe spins), so orbit shrinkage requires that the mutualinclination decreases. See Fabrycky et al. (2007) for adetailed description of this process. If coplanar triplesystems are found with i = 0 ◦ , they might also be inter-preted as resulting from fragmentation of a rather thin Fig. 8.—
Two dimensional histogram of the cosine of the finalmutual inclination of inner and outer binaries versus the final in-ner binary period. Contours are spaced in increments of 50, andthe gray-scale represents the number density with a finer grada-tion. Bin size is a quarter of an order of magnitude in P in,final and 0 . i final , which is the same resolution as the tickmarks. The striking paucity of systems with i final ≈ ◦ and10 d < P in,final < d is due to attrition: systems that startedwith these values have been removed by KCTF to smaller periods.Many of these systems inhabit the spikes of Figure 7b, which areclear enhancements in this plot. The region between the dashedlines corresponds to initial values of systems that may lose enoughangular momentum by Kozai cycles, and enough orbital energy bytidal friction, to circularize to final periods less than 10 d (see text). disk, but an observation of the purely retrograde case( i = 180 ◦ ) would seem to require this dissipation mech-anism. Our simulations yield equal numbers of systemswith i final ≃ ◦ and i final ≃ ◦ , but this is an artifactof our assumption of isotropic initial conditions, which isprobably not valid for such compact systems.Now we step back from these small periods to surveythe whole range of periods affected by KCTF. In Fig-ure 8 we plot the distribution of systems as a functionof both cos i final and P in,final . The spikes of Figure 7bare prominent at low periods, but there is another strik-ing feature: a deficit of long-period inner binaries with i ≈ ◦ . These systems have evolved to lower periodsby KCTF. Here it is clear that KCTF can remove a sig-nificant fraction of near-perpendicular systems to smallperiods, over a wide range of inner binary periods, from ∼
10 d to ∼ d. For comparison, consider an initiallycircular inner binary that undergoes eccentricity growthby the Kozai mechanism, then is tidally circularized from e in,max at constant orbital angular momentum. By thisprescription, the locus of initial systems that produce bi-naries with P in,final = 10 d is indicated by dashed lineson Figure 8. Systems between those lines may be ex-pected to evolve significantly by KCTF, if other sourcesof pericenter precession are negligible. The deficit oftriple systems with perpendicular orbits was identifiedby Harrington (1968) in the first theoretical paper onKCTF, and this effect should be tested by determiningthe mutual inclination distribution of triple stars. PERSISTENCE OF THE FINAL MUTUAL INCLINATION P out /P in , a circular inner orbit,and a mutual inclination that is often near the criticalone. Since the distinctive inclination distribution thatresults from KCTF (Fig. 7b) may ultimately provide thestrongest observational evidence for this process, it is im-portant to ask whether the inclination persists from theend of KCTF to the present. We point out three mecha-nisms for changing the mutual inclination but argue thateach has a negligible effect.First, the Galactic tide may cause the outer binaryto precess. The precession period for a binary with pe-riod P b is roughly P gal /P b , where P gal is the verticaloscillation period of the star in the Galactic disk, witha local value of [ π/ ( Gρ )] / ≈ yr[ ρ/ (0 . M ⊙ pc − )] / .For the precession period of the outer binary to be lessthan the age of the Galaxy, P ≈
10 Gyr, we require P b & P gal /P ≈ a out & AU. Pericenter precession of the in-ner binary due to such a distant third body will likely tobe overwhelmed by relativistic precession; equation (34)implies that τ ˙ ω GR | e in =0 must be . a in,initial is very large. For example, an inner bi-nary with an initially circular orbit can only satisfy thisconstraint if a in,initial &
25 AU, given a out = 10 AU.For an orbit with a in,initial = 75 AU, the pericenter dis-tance . .
03 AU required for substantial tidal friction re-quires a maximum eccentricity in the Kozai cycle givenby 1 − e max < . | i − ◦ | . ◦ ). Larger or smaller values of a in,initial have an even stricter requirement. Therefore the num-ber of systems for which KCTF yields a close binary,after which the Galactic tide reorients the outer binary,is negligibly small.Second, individual passing stars will perturb the outerorbit, changing its orbital elements. The timescale forchanging the outer binary’s angular momentum is simi-lar to the timescale of changing its energy, which corre-sponds to the disruption timescale. For the typical den-sity (0 . M ⊙ / pc ) and velocity dispersion ( ∼
40 km/s) ofstars in the disk, this timescale is shorter than 10 Gyronly for binary semi-major axes above ∼ × AU. Asabove, such distant third bodies will not produce short-period inner binaries via KCTF with any reasonable fre-quency, so mutual inclination change by passing stars isalso ineffective.Finally, let us specialize to a planet that has migratedby KCTF. Generally its orbit will not lie in the host star’sequatorial plane, since tidal dissipation in the star willbe too weak to align the stellar spin with the planetaryorbit (see § ∼ yrs.As the stellar spin precesses, the planet’s orbit must alsoprecess to conserve angular momentum. However, themagnitude of the associated change in i is . ◦ , sincethe orbit of a hot Jupiter usually has more angular mo-mentum than the spin of its host, so this effect is unlikelyto change the inclination distribution significantly. Thethick appearance of the line in the final state in Figure 1cis this type of oscillation. APPLICATION TO HOT JUPITERS
Stars are born by the fragmentation of molecularclouds. Planets, however, are believed to form in proto-stellar accretion disks, after material has settled arounda star. In binary star systems, it might be expectedthat the angular momentum of the disk around each starwould come into alignment with the angular momentumof the binary orbit. The star accretes high angular mo-mentum material from this orbit-aligned disk, so the spinof the host star of the planetary system would likely bein alignment with the companion orbit as well. Thus wemight expect that both stellar spin angular momenta,the stellar orbital angular momentum about the systembarycenter, and the planetary orbital angular momen-tum around its host star are all aligned if the binarysemi-major axis is not too large. Hale (1994) has mea-sured inclination to the line of sight of the spins of stars inbinaries by comparing the rotational period of starspotsto the v sin i values of rotationally-broadened lines. Ifthe stellar spins are aligned with each other, they willhave zero difference in inclination to the line of sight.The converse is not true, but the prevalence of align-ment can still be assessed statistically. Thus Hale (1994)inferred that binaries are spin-aligned for a . −
40 AU,but become randomly oriented for larger orbits. There-fore, for nearly the entire exoplanet sample in binaries,it is likely that the protoplanetary disk was not alignedwith the companion’s orbit. Therefore planets arisingfrom these non-aligned disks may have a high inclina-tion with respect to the binary companion, so KCTFmay cause substantial evolution to the orbital distribu-tions of planets in binary systems. An alternative, butrarer, mechanism to produce non-aligned planets in bi-nary star systems is to form a planetary system arounda single star, to which is later added a binary compan-ion through a dynamical interaction in the birth cluster(Pfahl & Muterspaugh 2006).In the past decade, about 200 giant planets have beendiscovered, of which about 20% have orbital periods shortenough so that tides are important. The high frequencyof close-in planets was a surprise, since planet forma-tion theory suggested that giant planets can only formbeyond several AU, consistent with the current structureof the solar system. Therefore, a migration mechanism isneeded to reduce the angular momentum of these plan-ets by a factor of ∼
10. The leading candidate is diskmigration, in which torques between the planet and theremnant protoplanetary nebula transfer angular momen-tum from the planet to the gas (Goldreich & Tremaine1980; Ward 1997). There is some statistical evidencethat planets orbiting close to one member of a wide bi-nary have different properties—and hence a different for-mation or migration history—from planets orbiting iso-lated stars. Among short-period radial velocity plan-ets (
P <
100 d), the most massive planets ( M p,min > a out .
300 AU) that KCTF may op-erate (Desidera & Barbieri 2007). These short-period,massive planets in binary stars also have lower eccen-tricities than short-period planets orbiting single stars(Eggenberger et al. 2004). For these systems, KCTFprovides an alternative migration mechanism to inter-ozai cycles with tidal friction 13actions with the protoplanetary nebula. The predictionsof the previous sections for the period and mutual incli-nation distributions should still apply in the planetarycase, and there is an additional prediction of KCTF re-garding the alignment of the orbital plane with the stellarequatorial plane (see § Extra precession due to other planets
In the solar system, apsidal precession is dominatedby the gravitational perturbations from other planets;in multi-planet systems that are hosted by one mem-ber of a binary star, precession from other planets willcompete with the precession due to the companion star(Holman et al. 1997). Wu & Murray (2003) showed thatthe Kozai mechanism could be suppressed by small, un-detected masses in the HD 80606 system, since its com-panion star is distant so the tidal field and precessionrate from it are small. Innanen et al. (1997) have inves-tigated the instability of the solar system’s giant plan-ets that would be caused by a companion star in an in-clined orbit, as a function of inclination and companionmass. Marzari et al. (2005) have investigated how a close( ∼
50 AU) binary companion, coplanar with the planets,affects their scattering. The Kozai mechanism, in sys-tems in which it operates, may also lead to planet-planetscattering (Malmberg et al. 2006).The key to why massive, short-period planets are pref-erentially found in binaries may lie in the competitionbetween mutual planetary precession and precession dueto the companion star, but this is a complicated processand the sign of the predictions is not clear.Take, as an example, an ensemble of planetary systemsin which the total mass of the planets is always the same,but the mass assigned to individual planets varies ran-domly. Consider a case in which one planet is substan-tially more massive than the others. Initially the mutualperturbations of the planets suppress Kozai oscillationsin all of them. Once moderate eccentricities and inclina-tions are built up in the planetary system, perhaps bymutual perturbations rather than the Kozai mechanism,the small planets become destabilized and are ejected.These ejected bodies no longer suppress Kozai cycles, sonow KCTF may cause migration for the largest planet.Since the Kozai cycles depended on the planet ejecting itsneighbors, this argument suggests that KCTF migrationis more effective for more massive planets.As a foil, consider an ensemble of planetary systems inwhich the total mass of the planets varies randomly, butis always divided into some number N of planets of equalmass. If the total planetary mass is small enough, Kozaicycles will proceed unhindered, and one or more planetsmay migrate by KCTF. The presence of the small plan-ets that failed to suppress Kozai cycles may not be de-tectable with radial velocity surveys, but the planet thatmigrated may be detected due to the larger reflex veloc-ity that it induces in the star. This argument suggeststhat KCTF migration is more effective for less massiveplanets.Since these two arguments yield opposite conclusions,at this time we can make no prediction regarding whetherKCTF migration favors high or low mass giant planets. The period distribution of planets in binaries
Fig. 9.—
The cumulative distributions of two populations ofplanets: those for which Kozai cycles may have been possible(Group 1) and those for which they are not (Group 2). Group 1consists of all extrasolar planets having a binary stellar companionbut no other detected planets; Group 2 consists of all other knownextrasolar planets. There is no statistically significant differencein the distributions, thus no statistical evidence for KCTF in theperiod distribution of planets. In contrast, to date the strongestevidence of KCTF in triple stars is the period distribution of innerbinaries (Tokovinin et al. 2006).
The period distribution produced by KCTF (Figure 5)is quite similar to the “pile-up” of hot Jupiters near 3 d(Ford & Rasio 2006). We have conducted the followingstatistical test to look for direct evidence of KCTF inthe period distribution of the exoplanet sample. Wesplit the radial velocity planets into two groups: (1)single planets in a system with multiple stars accordingto Raghavan et al. (2006), and (2) planets orbiting sin-gle stars and planets in multiple-planet systems. Group(1) contains the systems that could exhibit Kozai cycles,now or in the past, since a third body is known andthere are not other planets that could suppress Kozaicycles. Group (2) are the planets for which Kozai cyclesare less likely to be present (assuming that multiple plan-etary systems are roughly coplanar and that they sup-press Kozai cycles driven by a stellar companion). Theremay be systems with single planets and an undetected third body that causes Kozai cycles (Takeda & Rasio2005), but this possibility simply reduces the power—and should not bias—the statistical test. In Figure 9 weplot the cumulative distributions of planetary period forthese two groups. We employ a Kolomogorov-Smirnovtest to assess whether they are statistically indistinguish-able. The biggest gap between the distributions lies at P = 15 .
766 d, actually in the direction opposite from theexpectation of KCTF; it is D n = 0 . n = 27 and n = 163, the probability of having a gap at least thislarge is 72% (Press et al. 1992). Therefore, there is nostatistical evidence for KCTF in the period distributionof exoplanets. As noted earlier, Tokovinin et al. (2006)have employed the same test, finding different period dis-tributions for isolated binaries and binaries in triples witha probability of 0 . Downloaded on September 15, 2006 from http://vo.obspm.fr/exoplanetes/encyclo/catalog.php
Planetary inflation
The energy associated with a hot Jupiter’s orbit isabout 10 times its binding energy. If the planet hasa high eccentricity and small pericenter distance, thenits radius will inflate from tidal heating as the or-bit circularizes (Bodenheimer et al. 2001; Gu et al. 2003;Bodenheimer et al. 2003). For instance, for HD 80606b,which may be undergoing KCTF migration currently,Wu & Murray (2003) estimate a tidal luminosity of10 erg s − . For its minimum mass of 3 . ∼
9, so the planetary radius should beincluded in self-consistent integrations of KCTF migra-tion. Since the timescale for circularizing is a very strongfunction of the radius of the planet, the maximum semi-major axis out to which planetary orbits are circularizedwill be diagnostic of the planetary radius at migration.There are three massive ( > . . Li) are found in largeabundance, a constraint on when the planetary materialaccreted can be given (Israelian et al. 2001).
TABLE 1Expected alignment properties of most triples afterKCTF. system type: m - m - m alignment of: star-star-star star-planet-star m -spin / inner-orbit yes yes m -spin / inner-orbit yes noinner-orbit / outer-orbit no; i ≈ ◦ or 140 ◦ likely These effects may be tested for the planet HAT-P-1b(Bakos et al. 2007), a transiting planet in a binary stellarsystem. It could be in the final stages of circularizationafter KCTF migration, which would lead both to someresidual eccentricity and to an inflated radius; the latteris securely observed.
Spin-orbit alignment
In close binary star systems, including those broughtclose by KCTF, the spins of the stars are expected toalign with the orbital angular momentum in a timescaleshort compared to the circularization timescale. How-ever, a star’s spin probably does not align with the or-bital angular momentum of a hot Jupiter it hosts be-cause of the small mass of the planet (see Table 1). Ameasurement of ψ , the angle between the stellar spin an-gular momentum and the planet orbital momentum, cantherefore constrain the history of the system.It can be expected that if a planet migrated via in-teractions with the disk, it will remain in the same or-bital plane to within an angle much less than a radian.However, for KCTF migration, the angular momentumof the planet precesses about the angular momentum ofthe inclined companion. This precession happens gen-erally when a body in an inclined orbit is introduced.Even the coplanarity of the solar system (whose net or-bital angular momentum is dominated by Jupiter, and ψ = 7 ◦ ) provides constraints on unseen planets in in-clined orbits at large distances (Goldreich & Ward 1972).As tidal dissipation takes over during KCTF migration,the planet’s orbit eventually couples more strongly to itshost’s equatorial bulge than to the stellar companion,and misalignment persists as a fossil record of the earlierperiod of precession driven by the companion. There-fore, if planets are formed and migrate via disk torquesto become hot Jupiters on a timescale short comparedto the precession timescale due to the stellar compan-ion, then we may expect to see approximate spin-orbitalalignment even in binary systems. On the other hand,if there was a period in which the planet’s orbital evolu-tion was dominated by the binary companion, and laterit came to be a hot Jupiter through KCTF migration,then non-alignment between stellar spin and planetaryorbit will be the norm.We integrated a series of 1000 systems to illustrate thiseffect. Each system had identical parameters except for i initial , which ranged from 84 . ◦ to 90 ◦ (an evenly-spacedgrid of cos i initial between 0 and 0 . m = m = 1 M ⊙ , m = 10 − M ⊙ , and started with a in = 5 AU, e in = 0 .
1, and ω in = Ω in = 0 ◦ . Theouter binary had a out = 500 AU and e out = 0. Beforethe planet has migrated at all, the Kozai timescale forthese systems is τ initial = 2 .
36 Myr (eq. [1]). The hoststar’s spin period was set to 10 d, beginning with zeroobliquity ( ψ = 0). The planet was given a viscous timeozai cycles with tidal friction 15 t V = 0 .
01 yr (see appendix) and started with no spin.Once the orbit has shrunk to a in = 0 .
15 AU, planetaryorbital plane precession is dominated by the host star’sbulge rather than the companion, so ψ stops evolving;we call the time it takes to reach this point t a =0 . AU .The system with the smallest inclination migrated theslowest, with t a =0 . AU = 4 . i initial = 90 ◦ + δi has the same valueof ψ final as a system with i initial = 90 ◦ − δi , which isrequired by the symmetries of the equations of motionand the chosen initial conditions. Thus, the final distri-bution of ψ from this calculation is a prediction for hotJupiters that have migrated by KCTF after beginningwith isotropic inclination relative to a companion star.The results are given in Figure 10. Systems with i initial within 3 . ◦ of 90 ◦ , corresponding to 60% of the total,only undergo one Kozai cycle; i.e., e in attains a singlemaximum, after which a in and e in are damped. Such sys-tems precess a fraction of a radian, therefore they form anorderly sequence at the smallest periods ( P final < . ψ attained. First, duringmost of these cycles, the host star maintains its space ori-entation, as the torque from the planetary orbit is small.Second, tidal dissipation is only important when e in isnear its maximum, such that i is close to i min ≈ ◦ (Fig. 2). Together, these considerations imply the sys-tems will produce hot Jupiters whose host stars haveobliquities in the range i initial − i min . ψ . i initial + i min .This range approximately describes the results for plan-ets with P final between 3 d and 5 d. For P final & ψ to lower values. (We have verified that this finalpartial realignment does not occur if the planet’s massis 10 − M ⊙ ; its orbital angular momentum is then only ∼
1% of the stellar spin momentum, so the planet cannotsubstantially reorient the star.)So far, besides the solar system, there are only fourplanetary systems for which ψ has been constrained(see Table 2). The Rossiter-McLaughlin (RM) effect(Rossiter 1924; McLaughlin 1924; Gaudi & Winn 2007)constrains the sky-projection of ψ (called λ in the lit-erature) for planets that transit their host star. TheRM effect arises because the spectral lines of the starare rotationally broadened, and the planet blocks por-tions that are red-shifted or blue-shifted as a function oftime when it transits the stellar disk. The derived val-ues of the alignment angle are effectively sky-projected(hence underestimated), but they are of the same mag-nitude as the solar system and are inconsistent with thedistribution of ψ in Figure 10, suggesting that at most amodest fraction of hot Jupiters are produced by KCTF.A prime candidate for the measurement of the RM effectis the transiting planet HAT-P-1b (Bakos et al. 2007),for which misalignment would be corroborating evidencefor the KCTF migration hypothesis, made plausible byits inflated radius.Alternatively, the timing of migration can be stronglyconstrained by the precise alignment of the transitingplanet HD 189733b (Winn et al. 2006). According to Fig. 10.— (Top) The final period of planets after KCTF migra-tion versus the final stellar obliquity ψ (the angle between stellarspin and planetary orbit)—see § ∼ . a in = 5 AU to 0 .
15 AU, in units of the initialKozai timescale τ = 2 .
36 Myr (eq. [1]). (Bottom) Histogram of ψ . Misalignment is common, and even retrograde ( ψ > ◦ ) orbitsare possible. This misalignment is observable in transiting planetsthrough the Rossiter-McLaughlin effect and would provide strongevidence for KCTF migration of a hot Jupiter in a wide binarysystem. TABLE 2Measurements of the sky-projection of the angle betweenstellar spin and planetary orbit ( λ ) for transitingexoplanets. Of these, only HD 189733 is known to be astellar binary. Exoplanet λ ReferenceHD 209458b − . ◦ ± . ◦ Winn et al. (2005)HD 189733b − . ◦ ± . ◦ Winn et al. (2006)HD 149026b 11 ◦ ± ◦ Wolf et al. (2006)TrES-1b 30 ◦ ± ◦ Narita et al. (2007)
Bakos et al. (2006) there is an M-dwarf companion at aprojected distance of 219 AU, and preliminary measure-ments exclude coplanarity between the inner and outerorbits at the 4 − σ level. If the outer orbit’s misalign-ment is confirmed, it may be possible to constrain thetimescale of formation and disk migration to less thanabout τ ≈ DISCUSSION
We have described the evolution and final orbital ele-ment distribution of binary stars resulting from the com-bined effects of the gravitational influence of a tertiarycompanion star and tidal friction. In particular, weshow that Kozai cycles plus tidal friction (KCTF) canstrongly enhance the number of binary stars with periodsin the range 0.1 d to 10 d; briefly, the distant compan-ion induces strong eccentricity oscillations (Kozai cycles)in systems in which the inner and outer binary orbitalplanes are nearly perpendicular, and these eccentric or-bits are circularized near pericenter by tidal friction.The logarithmic period distribution of inner binariesproduced by KCTF shows a peak near 3 d and declinesfrom 3 d to 10 d. These features are consistent withthe empirical period distribution of inner binaries of hi-erarchical systems found by Tokovinin et al. (2006), andwe confirm these authors’ interpretation that KCTF isthe cause of this feature. Therefore KCTF appears toproduce most close binaries.The dominance of KCTF for producing close binariesmay be confirmed by testing our prediction of a modifiedmutual inclination distribution. We expect an enhance-ment of systems with mutual inclination close to the crit-ical values of 39 . ◦ or 140 . ◦ if P in is in the range 3-10 d.Interferometric measurements have been determining themutual inclination in triple stars with inner periods smallenough for tides to be important (Muterspaugh et al.2006), and with improvements in limiting magnitude andprecision they will be able to measure enough systems totest this theoretical prediction. Planetary inclination an-gles relative to a stellar companion will in some cases bemeasurable by SIM PlanetQuest .As the observations improve, the present theory shouldbe augmented to refine the predictions. In particular,most of the systems in the spikes in the inclination distri-bution (Fig. 7b) have 2 < G out /L in <
10, so our assump-tion that the outer binary dominates the angular momen-tum is only marginally valid. A more accurate model forthe known three-body gravitational dynamics may alsohave some effect on the period distribution produced byKCTF. For instance, Ford et al. (2000) showed that in-cluding octupole terms in the interaction Hamiltoniancauses Kozai cycles to be quasi-periodic, and some peri-center approaches can be much closer than those com-puted with quadrupole terms alone (their Fig. 5). Thesecloser approaches may lead to smaller final periods aftercircularization than we have predicted. Another aspectof the present theory that could be improved is modelingdynamical tides as well as, or instead of, the equilibriumtide. This more sophisticated tidal theory follows howthe tidal force excites free modes of oscillation in eachstar and how these modes are damped. It may be usefulin the present application, in which high eccentricities arecommon and the timescale of pericenter passage is com-parable to the periods of oscillation modes. However, thenatural first step to refine the present analysis is to im-prove the model of point-mass gravitational interactions(which are completely understood) before attempting toimprove the model of tidal interactions (which are quiteuncertain).We have investigated the period distribution that isproduced by KCTF alone and found that it preferentially produces detached binaries with P in ≃ contact binaries. KCTF and magnetic windshave often been regarded as competing theories of angu-lar momentum loss for close binaries. However, both aresimple additions of an extra star to well-grounded em-pirical relations: angular momentum loss by magneticwinds adds a binary companion to the spin-down pro-cess of single stars, and KCTF adds a third star to thecircularization process of close binary stars. Thereforewe believe both mechanisms must be operating at somelevel, and their relative contribution to the orbital evolu-tion of close binaries is an interesting subject for futurework.We have considered hierarchical triple systems in thefield of the Galaxy, where the space density of stars issmall enough that the triples are presumably dynami-cally isolated (see, for instance, § add athird body to primordial binaries (at least temporarily),in which case the uncorrelated orientations of inner andouter binaries will be above the critical inclination ∼ § § CONCLUSIONS
We conclude by reviewing the predictions of KCTF: (1)Period distributions for hot Jupiters and close binariesin systems with stellar companions have a peak at a fewdays and a minimum at longer periods ( ∼
10 d) for whichtidal circularization becomes ineffective. (2) For triplesin which the inner binary has a period between 3 and10 d, there is an enhancement of systems whose mutualinclination between inner and outer binaries is near 40 ◦ and 140 ◦ ; for triples with longer inner periods, there is apaucity of systems with nearly perpendicular orbits. (3) Hot Jupiters resulting from KCTF migration generallyhave orbital angular momentum misaligned with the stel-lar spin axis by large angles, frequently even larger than90 ◦ , and they may have orbits circularized to greater pe-riods than for hot Jupiters orbiting single stars due toinflation of the planetary radius during migration.The role of KCTF in the formation of close binary starsand hot Jupiters can be tested by measurements of thefollowing quantities in carefully defined samples of stars:(1) the period distribution of close binaries that havea tertiary component; (2) the mutual inclination distri-bution of triples, in order to detect the enhancement ofsystems near the critical angles (these measurements willnormally require interferometry); (3) the angle betweenstellar spin angular momentum and planetary orbital an-gular momentum for planets hosted by stellar binaries,using the RM effect, transit timing, or determination ofthe orbit of the outer binary for transiting planets; (4)differential measurements of metallicity in stars in binarysystems that host planets, to detect stellar pollution fromoverflow of the Roche lobe of the planet; (5) the distribu-tions of mass, eccentricity, and period for hot Jupiters.This research is supported by NASA awardNNG04H44G to ST. We thank Bohdan Paczy´nskifor initiating and supporting the project with greatenthusiasm and remarkable insight. We benefitedfrom discussions with P. Eggleton, M. Krumholz, A.Tokovinin, M. van Kerkwijk, and Y. Wu. APPENDIX
TERMS FOR THE EQUATIONS OF MOTION
Here we give the functional forms of the coefficients in the differential equations (3-6). V = 9 t F (cid:20) / e in + (15 / e in + (5 / e in (1 − e in ) / − h
18 ˙ l in / e in + (1 / e in (1 − e in ) (cid:21) , (A1) W = 1 t F (cid:20) / e in + (45 / e in + (5 / e in (1 − e in ) / − Ω h ˙ l in e in + (3 / e in (1 − e in ) (cid:21) , (A2) X = − m k R µ ˙ l in a in Ω h Ω e (1 − e in ) − Ω q l in t F / e in + (5 / e in (1 − e in ) , (A3) Y = − m k R µ ˙ l in a in Ω h Ω q (1 − e in ) + Ω e l in t F / e in + (1 / e in (1 − e in ) , (A4) Z = m k R µ ˙ l in a in " h − Ω q − Ω q − e in ) + 15 Gm a in / e in + (1 / e in (1 − e in ) . (A5)Here ˙ l in = 2 π/P in = ( G ( m + m ) /a in ) / is the mean motion. Similar expressions with subscripts 1 and 2 swappedhold for the second body. The parameters ( X, Y, Z ) form a vector in the ( ˆe in , ˆq in , ˆh in ) frame that gives its angularprecession rate relative to the inertial frame. Their relation to the orbital elements is (Eggleton et al. 1998): X = ˙ i cos ω in + ˙Ω in sin ω in sin i (A6) Y = − ˙ i sin ω in + ˙Ω in cos ω in sin i (A7) Z = ˙ ω in + ˙Ω in cos i. (A8)The dissipationless terms in these parameters can be found by computing ˙ i , ˙ ω in and ˙Ω in with the Hamiltonians of § m , the tidal-friction timescale is defined in terms of the viscous timescale t V (which we take to be8 Fabrycky and Tremainea constant): t F = t V (cid:18) a in R (cid:19) m ( m + m ) m (1 + 2 k ) − . (A9)Here, k is the classical apsidal motion constant, a measure of quadrupolar deformability which is related to the Lovenumber ( k L = 2 k ) and the coefficient Q E given by Eggleton & Kiseleva-Eggleton (2001): k = Q E / (1 − Q E ). We usethe typical value k = 0 .
014 valid for n = 3 polytropes when representing stars (Eggleton & Kiseleva-Eggleton 2001)and k = 0 .
25 valid for n = 1 polytropes when representing gas giant planets. Again, there is an analogous equationfor k . This viscosity causes the tidal bulge to lag the instantaneous direction of the companion by a constant time. Inthe Goldreich & Soter (1966) theory of tides, a quality factor Q is taken to be a constant. The average fraction of theenergy in the tide that is lost to frictional heat per radian of the orbit is Q − , and the tidal bulge lags the instantaneousdirection of the companion by an angle (2 Q ) − . When relating that angle to the time lag given by Eggleton et al.(1998), taking into account a stray factor of 2 mentioned in the appendix of Eggleton & Kiseleva-Eggleton (2001), wefind: Q = 43 k (1 + 2 k ) Gm R t V ˙ l in , (A10)assuming the dissipation is dominant in body 1. Taking t V to be a constant means Q ∝ P in ..