Survival of planets around shrinking stellar binariesDiego J. Muñoz1 and Dong Lai

Center for Space Research, Department of Astronomy, Cornell University, Ithaca, NY 14853

Edited by Neta A. Bahcall, Princeton University, Princeton, NJ, and approved May 29, 2015 (received for review March 23, 2015)

The discovery of transiting circumbinary planets by the Keplermissionsuggests that planets can form efficiently around binary stars. Noneof the stellar binaries currently known to host planets has a periodshorter than 7 d, despite the large number of eclipsing binaries foundin the Kepler target list with periods shorter than a few days. Thesecompact binaries are believed to have evolved fromwider orbits intotheir current configurations via the so-called Lidov–Kozai migrationmechanism, in which gravitational perturbations from a distant ter-tiary companion induce large-amplitude eccentricity oscillations inthe binary, followed by orbital decay and circularization due to tidaldissipation in the stars. Here we explore the orbital evolution ofplanets around binaries undergoing orbital decay by this mechanism.We show that planets may survive and becomemisaligned from theirhost binary, or may develop erratic behavior in eccentricity, resultingin their consumption by the stars or ejection from the system as thebinary decays. Our results suggest that circumbinary planets aroundcompact binaries could still exist, and we offer predictions as to whattheir orbital configurations should be like.

extrasolar planets | close binaries | celestial dynamics | N-body problem

To date, the Kepler spacecraft has discovered eight binary starsystems harboring 10 transiting circumbinary planets (1–8).

These systems have binary periods ranging from 7.5 d to ∼ 41 d,while the planet periods range from ∼ 50 d to ∼ 250 d. Remarkably,no transiting planets have been found around more-compact stellarbinaries, those with orbital periods of K 5 d. Planets around suchcompact binaries, if orbiting in near coplanarity, should havetransited several times over the lifetime of the Kepler mission.However, the shortest-period binary hosting a planet is Kepler-47(AB),with 7.44 d, despite the fact that nearly 50% of the eclipsingbinaries in the early quarters of Kepler data have periods shorterthan 3 d (9). Thus, the apparent absence of planets around short-period binaries is statistically significant (e.g., ref. 10).It is widely believed that short-period binaries (K 5 d) are not

primordial but have evolved from a wider configurations viaLidov–Kozai (LK) cycles (11, 12) with tidal friction (13–15). This“LK+tide” mechanism requires an external tertiary companionat high inclination to excite the inner binary eccentricity suchthat tidal dissipation becomes important at pericenter, eventuallyleading to orbital decay and circularization. A rough transition atan orbital period of 6 d has been identified as the separationbetween “primordial” and “tidally evolved” binaries (15). In-deed, binaries with periods shorter than this threshold are knownto have very high tertiary companion fractions (of up to 96% forperiods <3 d; see ref. 16), supporting the idea that three-bodyinteractions have played a major role in their formation.In synthetic population studies (15), stellar binaries with periods

shorter than ∼ 5 d evolved from binaries with original periodsof∼ 100 d. Interestingly, it is around binaries with periods of K 100 dthat transiting planets have been detected. It is thus plausible thatcurrent compact binaries with a tertiary companion may have oncebeen primordial hosts to planets like those detected by Kepler.In this work, we study the evolution and survival of planets

around stellar binaries undergoing orbital shrinkage via the LK+tide mechanism. We follow the secular evolution of the planetuntil binary circularization is reached and binary separation isshrunk by an order of magnitude. We show that the tertiarycompanion can play a major role in misaligning and/or destabi-lizing the planet as the binary shrinks.

A Planet Inside a Stellar TripleConsider a planet orbiting a circular stellar binary of total massMin =m0 +m1 and semimajor axis ain; the binary is a member ofa hierarchical triple, in which the binary and an outer companionof mass Mout orbit each other with a semimajor axis aout � ain.The secular (long-term) gravitational perturbations exerted on theplanetary orbit from the quadrupole potential associated with theinner binary and that from the outer companion cause the twovectors that determine the orbital properties of the planet, theangular momentum direction L̂p and the eccentricity vector ep, toevolve in time (if the inner binary has an equal-mass ratio and theouter companion has zero eccentricity, the octupole-order termsin the potential vanish exactly). The inner binary tends to make L̂pprecess around L̂in, the unit vector along the inner binary’s angularmomentum, at a rate approximately given by

Ωp‐in ≡12np

�μinMin

��ainap

�2

, [1]

where ap is the semimajor axis of the planet, np =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGMin=a3p

qis

the planet’s mean motion frequency (assumed to be on a circularorbit), and μin =m0m1=Min is the reduced mass of the inner stel-lar pair. Similarly, the outer companion of mass Mout tends tomake L̂p precess around L̂out at a rate approximately given by

Ωp‐out ≡ np

�Mout

Min

� 

�apaout

�3

[2]

(although we assume a circular outer companion here, theeccentricity of the outer orbit eout can be taken into account byreplacing aout with aout

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− e2out

p). In general, when the torques

from the inner binary and the outer companion are of com-parable magnitude, L̂p will precess around an intermediate vec-tor L̂p,eq, which corresponds to the equilibrium solution (i.e.,dL̂p=dt= 0) of the planet’s orbit under the two torques. For a

Significance

The detection of planets around binary stars (sometimes called“Tatooine planets”) in the last few years signified a major dis-covery in astronomy and posed a significant challenge to ourunderstanding of planet formation. So far, the discovered cir-cumbinary planets orbit relatively wide stellar binaries (with bi-nary orbital period greater than 7 d) and have their orbital axesaligned with the binary axes. The theoretical/numerical workreported in this paper suggests that there may be a new pop-ulation of circumbinary planets, which orbit around more-com-pact binaries (with periods less than a few days) and have theirorbital axes misaligned with the binary axes. Current observa-tional strategy inevitably misses this population of Tatooineplanets, but future observations may reveal their existence.

Author contributions: D.J.M. and D.L. designed research; D.J.M. performed research; D.J.M.contributed new reagents/analytic tools; D.J.M. analyzed data; and D.J.M. and D.L. wrotethe paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: dmunoz@astro.cornell.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1505671112/-/DCSupplemental.

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general mutual inclination angle iin‐out between the inner andouter orbits (where cos iin‐out = L̂in · L̂out), the equilibrium inclina-tion of the planet (the so-called “Laplace surface”; see refs. 17and 18), can be found as a function of its semimajor axis, forwhich L̂p,eq is always coplanar with L̂in and L̂out, with limitingstates corresponding to alignment with the inner binary (i.e.,L̂p,eq

��L̂in) at small ap, and alignment with the outer companion(i.e., L̂p,eq

��L̂out) at large ap. The transition between these twoorientations happens rapidly at the so-called “Laplace radius”rL, obtained by setting Ωp,out =Ωp,in, and is given by

rL =�

μin2Mout

a2in   a3out

�1=5

. [3]

Thus, there are three regimes (see Supporting Information fora schematic depiction) for the planet’s equilibrium orienta-tion: (regime a) Ωp‐in � Ωp‐out or binary-dominated regime(ap � rL); (regime b) Ωp‐out ≈Ωp‐in or transition regime (ap � rL);and (regime c) Ωp‐out � Ωp‐in or companion-dominated regime(ap � rL).In general, however, the vector L̂in is not fixed in space but

slowly precesses around L̂out, owing to the torque from the outercompanion (strictly speaking, both L̂in and L̂out precess around thetotal angular momentum vector of the system; however, for thehierarchical configurations presented here, the outer orbit containsmost of the angular momentum of the system, implying that L̂out isapproximately fixed in space). This means that the plane normal toL̂in × L̂out, where the equilibrium orientation vector L̂p,eq lives, isslowly rotating (see Supporting Information). This rotation rate is oforder

Ωin‐out ≡ nin

�Mout

Min

� 

�ainaout

�3

, [4]

where nin =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGMin=a3in

qis the mean motion of the inner binary.

Note that Ωp‐out=Ωin‐out = ðap=ainÞ3=2 � 1 in the companion-dominated regime, and Ωp‐in=Ωin‐out = ðΩp‐in=Ωp‐outÞðap=ainÞ3=2 =ðrL=apÞ5ðap=ainÞ3=2 � 1 in the binary-dominated regime. Thismeans that the precession of L̂in is always slow enough for L̂pto adiabatically follow. In other words, the classical Laplaceequilibrium formalism remains valid in the frame corotatingwith L̂in, and the three vectors L̂in, L̂out, and L̂p,eq remain co-planar at all times. Because the evolution is adiabatic, if L̂p startsparallel to L̂p,eq, it will remain parallel to the evolving L̂p,eq at latertimes, provided that this equilibrium orientation is a stablesolution (17).As studied by ref. 17, when iin‐out > 69°, circular orbits on the

Laplace surface are unstable to linear perturbations in theplanet’s eccentricity vector ep vector for a range of ap around rL.This instability manifests itself as an exponential growth of ep,until nonlinear effects come into play, resulting in erratic be-havior in both inclination and eccentricity. This means thatabove this critical value of iin‐out, planets cannot be placed atap ≈ rL, because the resulting high eccentricities could bring themtoo close to the binary, at which point they may collide with thecentral stars or be ejected from the system (e.g., ref. 19).Now consider what will happen to the planet’s orbit as the

inner binary undergoes orbital decay. For simplicity, let us as-sume that the binary remains circular during this process, andthat the angle iin‐out remains unchanged. Because orbital decaytakes place over a time scale tdecay much longer than the otherrelevant time scales (1=Ωp‐in, 1=Ωp‐out and 1=Ωin‐out), the systemwill evolve adiabatically. Thus, if the planet initially resides in thebinary-dominated regime (Ωp‐in � Ωp‐out, ap � rL), and lives onthe Laplace surface (L̂p

��L̂p,eq), it will transition to the compan-ion-dominated regime (Ωp‐in � Ωp‐out, ap � rL) through the in-termediate stage (Ωp‐in ≈Ωp‐out), as the inner binary’s semimajoraxis ain decreases. For a given value of ap, the transition occurswhen ain passes through a critical (Laplace) value,

ain,L ≡ 0.017�Mout

4μin

�1=2� ap1AU

�5=2� aout30AU

�−3=2AU [5]

obtained by replacing rL = ap in Eq. 3 and solving for ain. If thetransition region (ap ≈ rL) is stable, we expect the planet’s orbit toevolve smoothly following the Laplace surface (i.e., regime a→regime b→ regime c). For iin‐out > 69°, however, the planet willencounter an instability when ap ≈ rL, and may undergo erratic evo-lution, which may result in the planet being destroyed or ejected.In the LK+tide scenario for the formation of compact binaries,

the final inner binary separation ain,f depends on the properties ofthe outer companion (Mout, aout, and the initial inclination iin‐out)as well as on the short-range force effects between the inner binarymembers (15). Thus, for a given stellar triple configuration, theinner binary may or may not reach down to ain,L, depending on thevalue of ap (Fig. 1). If ain,f > ain,L, or equivalently, if

ap < 1.26�Mout

4μin

�−1=5� aout30AU

�3=5� ain,f0.03AU

�2=5AU, [6]

the planet will never cross the intermediate regime (ap ≈ rL), andit will thus remain “safe” (stable), regardless of the inclinationiin‐out, surviving the orbital decay of its host binary.

Evolution of Planetary Orbits Around Binaries UndergoingLidov–Kozai Cycles with Tidal FrictionThe greatest caveat to the application of classical Laplace equi-librium is that the inner binary does not remain circular during

Fig. 1. The three relevant precession frequencies (Ωp‐in, Ωp‐out, and Ωin‐out) as afunction of the shrinking binary semimajor axis ain. The binary starts at semi-major axis ain,0 = 0.3 AU and circularizes at ain,f = 0.024 AU (vertical black line).The other parameters are Min =Mout = 1M⊙, μin = 0.25, aout = 30 AU, andeout = 0. (Top) The case with ap =2 AU, and (Bottom) the case with ap = 1 AU. InTop, ain crosses ain,L = 0.097 AU (thick vertical gray line) during orbital decay,then the planet transitions from the binary-dominated regime into the com-panion-dominated regime. In Bottom, ain,f > ain,L = 0.017 AU, and the planetwill stay in the binary-dominated regime throughout the binary orbital decay.Note that in this example, ap = 1 AU is very close to the initial binary, anddynamical instabilities (not captured by secular calculations) might make thesurvival of these planets difficult during the early Lidov−Kozai cycles of the binary.

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orbital decay. Indeed, in the LK+tide mechanism (14, 15), theinner binary exhibits large oscillations in inclination and eccen-tricity under the influence of the external stellar companion. Thus,the binary axis L̂in not only precesses around L̂out but also un-dergoes nutation. The variation of the inner binary’s eccentricityvector ein also affects the torque on the circumbinary planet.To track the evolution of the planet’s orbit during the LK

oscillations and orbital decay of the inner binary, we solve nu-merically the secular equations of the planet’s eccentricity vectorep and angular momentum vector axis L̂p (Supporting Informa-tion), along with the evolution equations of the stellar triple (thesecular equations of motion govern the evolution of the orbitalelements instead of the position and velocity of individualbodies). We use the formalism of ref. 20 to follow the innerbinary’s orbit and parametrize the stellar tidal dissipation rateusing the weak friction model with constant tidal lag time. In thefollowing, we focus on a few representative examples and discussthe general behavior for the evolution of the four-body system.Fig. 2 depicts a system where the stellar triple has parameters

m0 =m1 = 0.5M⊙, Mout = 1M⊙, aout = 18 AU, and eout = 0 and ini-tial values ain,0 = 0.3 AU and iin‐out,0 = 73°, and where the planet is

initialized on a circular orbit at ap = 1.5 AU with L̂p aligned withL̂in. The parameters for the inner binary and the planet are chosento roughly correspond to the discovered Kepler systems. The pa-rameters for the outer orbit are chosen to ensure that LK cycles arenot suppressed by short-range forces and to guarantee the efficientorbital decay of the inner binary (15). In our calculations, theoctupole term in the potential has been ignored in the evolutionequations of the planet and the inner binary, a justified simplifi-cation because m0 =m1 and eout = 0. The inner binary experiencesLK oscillations and circularizes within a Hubble time provided thatenough tidal dissipation is present in the stars. The final (circu-larization) semimajor axis is ain,f = 0.053 AU (corresponding toan orbital period of 4.5 d). In this example, the planet initiallyresides in the binary-dominated regime, with Ωp‐in=Ωp‐out ≈65ðap=AUÞ−5 ≈ 8.6, and ap=rL,0 ≈ 0.65. After the inner binary hascircularized, the planet lies in the companion-dominated regime,with Ωp‐in=Ωp‐out ≈ 0.27 and ap=rL,f ≈ 1.3. We see that the planetremains on a circular orbit throughout its entire evolution, despitethe large variations in ein during the LK cycles. The longitude ofnodes of the planet (not shown in the figure) closely follows that ofthe inner binary during the early LK cycles and after circulariza-tion, implying that, for a large fraction of the time, L̂p is coplanarwith L̂in and L̂out. The planet’s inclination ip‐out also follows thebinary inclination iin‐out during the early stage of the LK cycles (Fig.2, Bottom Middle), but it decouples from the inner binary after ainhas started decreasing. At the end of the integration, when the bi-nary has circularized, the binary and planet are misaligned by 32°(Fig. 2, Bottom) and the planet inclination has settled onto asteady-state value. This final value, ip‐out ’ 14°, agrees with theequilibrium value of the end-state Laplace surface (withain,f = 0.053 AU and iin‐out,f = 46°) evaluated at ap = 1.5 AU.We have carried out calculations for a range of values of ap for

the same stellar triple configuration of Fig. 2. The results ofthese calculations are summarized in Fig. 3, which shows the

Fig. 3. Classical Laplace equilibrium surface (valid for ein = 0) at the begin-ning (red curve) and after circularization of the inner binary (orange curve)for the triple configuration of Fig. 2 (m0 =m1 = 0.5M⊙, Mout = 1M⊙, ain,0 = 0.3AU, iin‐out,0 = 73°, ain,f = 0.053 AU, iin‐out,f = 46.1°, and aout = 18 AU). The dot-ted portion of the red line indicates the range of radii at which the equi-librium surface is unstable (17). The final Laplace surface is stable for all apbecause ip‐out,f < 60°. The vertical dash-dotted lines indicate the Laplace radiiat the beginning (rL,0) and end (rL,f) of the binary orbital evolution. For thedifferent values of ap, vertical arrows connect the initial and final states,representing the evolution of the planet’s inclination obtained from thenumerical calculations. In each case, the planet orientation is initially alignedwith the local Laplace surface (or approximately aligned with the binary forap K 1.5 AU); after the inner binary has decayed and circularized, the planetinclination settles into a value coincident with final Laplace surface. Notethat, for illustrative purposes, we include values of ap down to 1 AU; how-ever, dynamical stability dictates that only planets outside ap ≈ 4ain ≈ 1.2 AU[when ein≈ 1 (19),] should survive the early LK cycles of the inner binary.

Fig. 2. An example of the coupled evolution of an inner binary within astellar triple (m0 =m1 = 0.5M⊙, Mout = 1M⊙, aout = 18 AU, and initial ain,0 = 0.3AU and iin‐out,0 = 73°) plus a planet with semimajor axis ap = 1.5 AU. Thedifferent panels show (Top) binary semimajor axis ain (green) and planetsemimajor axis ap (blue) and the Laplace radius rL (cyan); (Top Middle) ec-centricity of the binary ein (green) and eccentricity of the planer ep (blue);(BottomMiddle) inclination of the binary iin‐out (green) and inclination of theplanet ip‐out (blue) with respect to the outer companion; and (Bottom) mu-tual inclination between the planet and the binary ip‐in. The eccentricity andinclination of the binary exhibit LK cycles for about 10% of the integrationtime, until short-range forces arrest these oscillations (freezing ein at highvalues), at which point a slow phase of orbital decay takes place. The planetstarts in the binary-dominated regime (ap=rL,0 = 0.65). Its inclination ip‐outfollows closely that of the inner binary iin‐out until rL crosses ap (note that thedefinition of rL in Eq. 3 does not take into account the eccentricity of theinner binary ein), at which point these two inclination angles decouple from eachother. The planet ends in the companion-dominated regime (ap=rL,f = 1.3),and its inclination with respect to the binary ip‐in eventually settles into aconstant value of ∼ 32°.

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Laplace surfaces at the beginning and at the end of the evolu-tion, when the inner binary is circular. In each case, the planet isinitially aligned with the equilibrium orientation L̂p,eq, which is innear alignment with the inner binary for ap K 1.5 AU. We findthat the planet’s inclination evolves smoothly for all these casesas the binary experiences LK oscillations and orbital decay.Despite the complexity of the “intermediate” states, in which thebinary develops large eccentricities and the standard Laplaceequilibrium is not well defined, we find that in the end, theplanet’s inclination always lands on the final Laplace surface.Thus, these planets survive the orbital decay of the inner binarybut become inclined with respect to it by an angle given byip‐in,f = iin‐out,f − ip‐out,f (with iin‐out,f ≈ 46° for the parameters adoptedin Figs. 2 and 3), where ip‐out,f matches the equilibrium inclinationof the final Laplace surface. Because the Laplace equilibriuminclination angle decreases with increasing ap, we predict that theangle ip‐in of the planets that survive will increase monotonicallywith increasing ap.As noted before, when the mutual inclination iin‐out between

the inner circular binary and the external companion is greaterthan 69°, a portion of the Laplace surface is unstable (17). Inprinciple, a circumbinary planet may suffer a similar instability asa binary with large initial iin‐out undergoes LK+tide orbital decay.In Fig. 4, we show two examples (ap = 1.2AU and ap = 1.8AU forFig. 4, Left and Fig. 4, Right, respectively) of planets within astellar triple with aout = 30 AU, ain,0 = 0.3 AU, and iin‐out,0 = 83°(the other parameters are the same as in Fig. 2). At this initialinclination, the inner binary attains very high eccentricities andcan circularize very efficiently (alternatively, it requires rela-tively small tidal dissipation in the stars to circularize withina Hubble time). The final binary separation is ain,f = 2.55× 10−2AU (period of 1.5 d), and the inclination angle freezes outat iin‐out,f = 65.3°. The behavior of the planets is markedly dif-ferent from the one depicted in Fig. 2. For a planet located at

ap = 1.2 AU (Fig. 4, Left), the inclination angle ip‐out does notevolve smoothly as the inner binary decays but suffers a jump asrL crosses ap, subsequently oscillating around a reference angle.Moreover, the orbital eccentricity rapidly grows until it startsoscillating around a mean value of hepi≈ 0.16, maintaining fromthen on a steady-state behavior. For a planet at ap = 1.8 AU (Fig.4, Right), the orbital evolution is even more complex. In this case,the exponential growth in eccentricity does not saturate at amoderate value. Instead, ep reaches values close to 1. The erraticevolution in ep is accompanied by a similar behavior in theplanet’s inclination ip‐out. Instead of oscillating around a mean(equilibrium) value, ip‐out covers the entire range ð0°, 180°Þ. Thehigh planet eccentricities reached in this case make it very un-likely for the planet to survive the orbital decay of the innerbinary. Such high eccentricities will inevitably bring the planettoo close to the inner binary, a region that is known to be un-stable (19, 21). In this case, ejections from the system or physicalcollisions with the central stars are to be expected.In Fig. 5, we show the initial and final inclinations (Fig. 5,

Top), and the respective final eccentricities (Fig. 5, Bottom),computed for a set of values of ap using the same stellar tripleconfiguration of Fig. 4. As in Fig. 3, these results are showntogether with the Laplace equilibrium surface solutions for theinitial state ðain,0 = 0.3 AU  and  iin‐out,0 = 83°Þ and the final stateðain,f = 0.026 AU  and  iin‐out,f = 65.3°Þ. Unlike Fig. 3, we find that,depending on ap, planet orbits do not always stay circular, andtheir inclinations ip‐out do not always land exactly on the finalLaplace surface. For ap K rL,f , planets end up very close (onaverage) to the final Laplace surface (while exhibiting someminor oscillations around it), and maintain a negligible eccen-tricity. For ap J rL,f , planets suffer a small kick in eccentricityas they cross the “transition” regime ðap = rLÞ, and their inclinat-ions oscillate with significant amplitude around a mean value thatis close, but not necessarily equal to, the one given by Laplace

Fig. 4. Similar to Fig. 2, but for a triple system that is initialized at a higher inclination iin‐out,0 = 83∘. The other parameters for the tare ain,0 = 0.3 AU andaout = 30 AU, with the same stellar masses as in Fig. 2. Two examples are shown: ap = 1.2 AU (Left) and ap = 1.8 AU (Right), both exhibiting quite differentplanetary evolution compared with Fig. 2. In the case of ap = 1.2 AU, the planet starts with ap=rL,0 = 0.38 and ends with ap=rL,f = 1.02; in the case of ap = 1.8 AU,the planet starts with ap=rL,0 = 0.57 and ends with ap=rL,f = 1.54.

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equilibrium (see Fig. 4, Left). At even larger apðJ 1.5 AUÞ,we find that the evolution of the planet is no longer regular(see Fig. 4, Right): Both ep and ip‐out undergo large-amplitude,erratic variations (ep ’ 0 - 1 and Ip‐out ’ 0°‐80°). Indeed, forlarge values of ap, the planet’s evolution is most likely chaotic,because the results depend sensitively on the initial conditions(Supporting Information). Erratic evolution (in eccentricity andinclination) may last indefinitely or may end before circulari-zation of the inner binary has completed, in which case planetscan exit the erratic phase at a random inclination (includingangles > 90°). In either case, these planets, having experiencederratic, large-amplitude variations of ep, are likely to be ejectedfrom the system or to collide with the binary stars.In the above, our calculations have ignored the mass of the

circumbinary planet mp based on the assumption that the plan-etary mass is always much smaller than Min and Mout. However,over secular time scales, a finite planet mass can affectthe dynamics of the inner binary to the point of suppressingthe eccentricity oscillations caused by the tertiary (22). Theplanet-induced precession frequency of the binary is of orderΩin‐p ’ ninðmp=MinÞ  ðain=apÞ3. The condition Ωin‐p ’ Ωin‐out al-lows us to define the critical planet mass

mp,crit ’ 0.13MJ

�Mout

1M⊙

�� ap1.5AU

�3� aout30AU

�−3[7]

(where MJ is the mass of Jupiter) above which the precession ofthe binary due to the planet is faster that due to the tertiary star.For small mp, the effects of a finite planet mass on the LK cyclesare qualitatively similar to those of other short-range forces (15,

23), imposing an upper limit on the maximum eccentricity of thebinary. In Fig. 6, we show the maximum eccentricity achievedby the inner binary as a function of planet mass obtained fromintegrations of the four-body secular system (see SupportingInformation). For the example depicted in Fig. 2 (ap = 1.5AU and mp,crit ’ 0.6MJ), we find that mp J 1MJ ’ 1.7mp,crit isenough to substantially suppress the oscillations in ein. Formp K 0.3MJ ’ 0.5mp,crit (about the mass of Saturn), the minimumpericenter separation of the binary ain,0ð1− ein,maxÞ≈ 0.09ain,0 isonly 17% larger than 0.077ain,0, the value corresponding tomp = 0. Such a planet (mp K 0.3MJ) will only delay the orbitalshrinkage of the inner binary, but not prevent it (see SupportingInformation for an example).Throughout this paper, we have included only the quadrupole

potential from the tertiary companion acting on the inner binaryand the planet. This is a good approximation when the com-panion has zero orbital eccentricity. For general companion ec-centricities, octupole and higher-order potentials may introducemore complex dynamical behaviors for the inner binary and forthe planet (see, e.g., refs. 23–26). For example, in N-body cal-culations (which include high-order terms automatically), theplanet may attain a nonzero eccentricity as the inner binarydecays even in the moderate inclination case (see SupportingInformation for one such example). A systematic study of thesecomplex “high-order” effects is beyond the scope of this paperand will be the subject of future work.

DiscussionWe have explored the orbital evolution of planets around bi-naries undergoing orbital decay via the LK+Tide mechanismdriven by distant tertiary companions. We have shown thatplanets may survive the orbital decay of the binary for tertiarycompanions at moderate initial inclinations (iin‐out,0 K 75°). Insuch cases, planets on circular orbits adiabatically follow anequilibrium solution as the triple system evolves, becomingmisaligned with their host binary; the final misalignment angleip‐in is a monotonically increasing function of the binary-planet

Fig. 5. (Top) Similar to Fig. 3, but for the triple configuration shown in Fig. 4(ain,0 = 0.3 AU, iin‐out,0 = 83°, ain,f = 0.026 AU, iin‐out,f = 65.3°). (Bottom) Thecorresponding final eccentricity of each planet. Error bars specify the oscil-lation amplitude around the mean value. Wide blue bands (for ap J1.5 AU)denote planet orbits with erratic behavior in inclination, covering the entirerange ½0°, 180°Þ, at any point during their evolution. Note that, for somevalues of ap, planets do reach regular values in eccentricity and inclinationeven after having experienced erratic evolution during a finite period beforebinary circularization; such cases are still depicted by blue bands, becausetheir survival is deemed unlikely (see Supporting Information).

Fig. 6. (Top) Maximum eccentricity ein,max of the inner binary in the tripleconfiguration of Fig. 3 achieved during the LK cycles as a function of planetmass mp for three different values of the planet semimajor axis ap: 1.2 AU(blue), 1.5 AU (red), and 1.8 AU (orange). Vertical lines denote the value ofmp,crit (Eq. 7) for each of the different values of ap. (Bottom) Same as Top,but for a triple configuration as in Fig. 5. In general, LK oscillations are en-tirely suppressed for mp J 2mp,crit. For smaller planet mass (mp < 1

2mp,crit), theeccentricity oscillation amplitude is only slightly modified.

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distance ap. At higher inclinations ðiin‐out,0 J 80°Þ, the adiabaticevolution is broken when planets encounter an unstable equi-librium. Then the planet orbit can develop erratic behavior ineccentricity and inclination. Very eccentric circumbinary orbitsmay be disrupted by the inner binary via dynamical instabilities,resulting in either the ejection of the planet or its collision ontothe stars. Interestingly, even in this high-inclination regime, wehave found that some planets may evolve into stable, misalignedand eccentric orbits.In our scenario, the abundance of misaligned planets around

compact binaries depends on the frequency of moderate initialinclination stellar triples relative to those with high inclinations.High-inclination stellar triples may be the progenitors of themajority of compact binaries, because the very high eccentricitiesreached by the inner binary make orbital decay faster. Our cal-culations suggest that planets within such high-inclination tripleshave less chances of survival during the inner binary’s orbitaldecay. The efficiency of tidal decay depends on the dissipationtime scale tV within the stars (see Supporting Information). Wehave found that dissipation time scales of order 20− 50 y cancircularize inner binaries with iin‐out,0 J 78° within a Hubble time,but if iin‐out,0 ∼ 70°, then tV ’ 1−5 y is required. However, giventhe large parameter space in orbital configurations, and theuncertainty in realistic values of tV (which may vary during stellarevolution), we cannot discard the possibility that some binaries,perhaps still undergoing orbital decay and circularization, may bepart of moderate-inclination stellar triples, and may therefore becandidate hosts to highly misaligned planets.An additional caveat to the abundance of misaligned circum-

binary planets that is not addressed in this work concerns thelikelihood of planets forming within inclined hierarchical tripleswith aout=ain ≈ 100. Planet formation will be limited by disk trun-cation from inside (at a≈ 3ain) and from outside (at a∼ aout=3)(27, 28). Thus, for the parameters explored in this paper, planetswould be confined to form between 1 AU and 10 AU. In additionto disk truncation and warping (29), planetesimal dynamics in thissystems could be affected by the tidal forcing of the inner binary

and the outer companion, introducing additional complications tothe formation of planetary cores (30, 31).As noted before, currently, no planets have been detected

around eclipsing compact (Pin K 5 d) stellar binaries. Our worksuggests that if planets are able to form within (moderately)compact triples, they are likely to survive the tidal shrinkage ofthe central binary, evolving into inclined orbits. The detection ofthese misaligned circumbinary planets may be challenging. Theplanets that survive the orbital decay of the binary lie close to/onthe Laplace surface, which follows with the precession of theinner binary axis L̂in with respect to the outer binary axis L̂out.The coupled precession of the inner binary and the planet or-bits will produce short-lived “transiting windows,” but these win-dows appear periodically over very long time scales [of order1=Ωin‐out ≈Pinðaout=ainÞ3 ≈ 105 − 106 y]. An alternative detectionstrategy is to look for eclipse timing variations. The perturbationon the inner binary exerted by a planet of mass mp introduces atiming signature (on the time scale of the planet’s orbital period)of magnitude ∼ ð3=8πÞPinðmp=MinÞðain=apÞ3=2 (32, 33). For val-ues of mp ≈ 0.5MJ ≈ 0.0005Min, ap ≈ 1.0 AU, ain ≈ 0.05 AU, andPin ≈ 5 d, the maximum eclipse timing variation is of order ∼ 0.3 s,approaching the noise level for some nearby binaries but, ingeneral, still below the detection limits for most eclipse timingdetections (34). However, short cadence data with current ob-servational capabilities might provide enough timing precision toaccomplish such measurements.

Note.During of the revision of our manuscript, we became awareof a preprint by D. Martin, T. Mazeh, and D. Fabrycky, whichaddresses a similar issue (i.e., the dearth of planets around com-pact binaries) as our paper (35).

ACKNOWLEDGMENTS. We thank Sarah Ballard, Konstantin Batygin, MatthewHolman, and Bin Liu for discussions and comments. We also thank the referee,Daniel Fabrycky, for valuable comments and suggestions. This work has beensupported in part by National Science Foundation Grant AST-1211061 andNASA Grants NNX14AG94G and NNX14AP31G.

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