Survival of around shrinking stellar binaries

Diego 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 Kepler mission A Inside a Stellar Triple suggests that planets can form efficiently around binary . None Consider a planet orbiting a circular stellar binary of total mass of the stellar binaries currently known to host planets has a period Min = m0 + m1 and semimajor axis ain; the binary is a member of shorter than 7 d, despite the large number of eclipsing binaries found a hierarchical triple, in which the binary and an outer companion in the Kepler target list with periods shorter than a few days. These of mass Mout each other with a semimajor axis aout ain. compact binaries are believed to have evolved from wider into The secular (long-term) gravitational perturbations exerted on the their current configurations via the so-called Lidov–Kozai migration planetary orbit from the quadrupole potential associated with the mechanism, in which gravitational perturbations from a distant ter- inner binary and that from the outer companion cause the two tiary companion induce large-amplitude eccentricity oscillations in vectors that determine the orbital properties of the planet, the ^ the binary, followed by orbital decay and circularization due to tidal angular momentum direction Lp and the eccentricity vector ep,to dissipation in the stars. Here we explore the orbital evolution of evolve in time (if the inner binary has an equal- and the outer companion has zero eccentricity, the octupole-order terms planets around binaries undergoing orbital decay by this mechanism. ^ We show that planets may survive and become misaligned from their in the potential vanish exactly). The inner binary tends to make Lp L^ ’ host binary, or may develop erratic behavior in eccentricity, resulting precess around in, the unit vector along the inner binary sangular in their consumption by the stars or ejection from the system as the momentum, at a rate approximately given by binary decays. Our results suggest that circumbinary planets around μ 2 compact binaries could still exist, and we offer predictions as to what 1 in ain Ωp‐in ≡ np , [1] their orbital configurations should be like. 2 Min ap qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = G = 3 extrasolar planets | close binaries | celestial dynamics | N-body problem where ap is the semimajor axis of the planet, np Min ap is the planet’s mean motion frequency (assumed to be on a circular μ = = o date, the Kepler spacecraft has discovered eight binary orbit), and in m0m1 Min is the reduced mass of the inner stel- – lar pair. Similarly, the outer companion of mass Mout tends to Tsystems harboring 10 transiting circumbinary planets (1 8). L^ L^ These systems have binary periods ranging from 7.5 d to ∼ 41 d, make p precess around out at a rate approximately given by while the planet periods range from ∼ 50 d to ∼ 250 d. Remarkably, 3 Mout ap no transiting planets have been found around more-compact stellar Ω ‐ ≡ n [2] p out p M a binaries, those with orbital periods of K 5 d. Planets around such in out compact binaries, if orbiting in near coplanarity, should have (although we assume a circular outer companion here, the transited several times over the lifetime of the Kepler mission. eccentricity of the outerp orbitffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieout can be taken into account by However, the shortest-period binary hosting a planet is Kepler-47(AB), − 2 replacing aout with aout 1 eout ). In general, when the torques with 7.44 d, despite the fact that nearly 50% of the eclipsing from the inner binary and the outer companion are of com- binaries in the early quarters of Kepler data have periods shorter parable magnitude, L^ will precess around an intermediate vec- than 3 d (9). Thus, the apparent absence of planets around short- ^ p tor Lp,eq, which corresponds to the equilibrium solution (i.e., period binaries is statistically significant (e.g., ref. 10). dL^ =dt = 0) of the planet’s orbit under the two torques. For a It is widely believed that short-period binaries (K 5 d) are not p primordial but have evolved from a wider configurations via Lidov–Kozai (LK) cycles (11, 12) with tidal friction (13–15). This Significance “LK+tide” mechanism requires an external tertiary companion at high inclination to excite the inner binary eccentricity such The detection of planets around binary stars (sometimes called that tidal dissipation becomes important at pericenter, eventually “Tatooine planets”) in the last few years signified a major dis- leading to orbital decay and circularization. A rough transition at covery in astronomy and posed a significant challenge to our an of 6 d has been identified as the separation understanding of planet formation. So far, the discovered cir- between “primordial” and “tidally evolved” binaries (15). In- cumbinary planets orbit relatively wide stellar binaries (with bi- deed, binaries with periods shorter than this threshold are known nary orbital period greater than 7 d) and have their orbital axes to have very high tertiary companion fractions (of up to 96% for aligned with the binary axes. The theoretical/numerical work periods <3 d; see ref. 16), supporting the idea that three-body reported in this paper suggests that there may be a new pop- interactions have played a major role in their formation. ulation of circumbinary planets, which orbit around more-com- In synthetic population studies (15), stellar binaries with periods pact binaries (with periods less than a few days) and have their shorter than ∼ 5 d evolved from binaries with original periods orbital axes misaligned with the binary axes. Current observa- of ∼ 100 d. Interestingly, it is around binaries with periods of K 100 d tional strategy inevitably misses this population of Tatooine that transiting planets have been detected. It is thus plausible that planets, but future observations may reveal their existence. current compact binaries with a tertiary companion may have once been primordial hosts to planets like those detected by Kepler. Author contributions: D.J.M. and D.L. designed research; D.J.M. performed research; D.J.M. In this work, we study the evolution and survival of planets contributed new reagents/analytic tools; D.J.M. analyzed data; and D.J.M. and D.L. wrote around stellar binaries undergoing orbital shrinkage via the LK+ the paper. tide mechanism. We follow the secular evolution of the planet The authors declare no conflict of interest. until binary circularization is reached and binary separation is This article is a PNAS Direct Submission. shrunk by an order of magnitude. We show that the tertiary 1To whom correspondence should be addressed. Email: [email protected]. companion can play a major role in misaligning and/or destabi- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. lizing the planet as the binary shrinks. 1073/pnas.1505671112/-/DCSupplemental.

9264–9269 | PNAS | July 28, 2015 | vol. 112 | no. 30 www.pnas.org/cgi/doi/10.1073/pnas.1505671112 Downloaded by guest on September 29, 2021 general mutual inclination angle iin‐out between the inner and =     ^ ^ M 1 2 a 5=2 a −3=2 outer orbits (where cos iin‐out = Lin · Lout), the equilibrium inclina- ≡ out p out [5] ain,L 0.017 μ AU tion of the planet (the so-called “Laplace surface”; see refs. 17 4 in 1AU 30AU and 18), can be found as a function of its semimajor axis, for ^ ^ ^ = 3 which Lp,eq is always coplanar with Lin and Lout, with limiting obtained by replacing rL ap in Eq. and solving for ain.Ifthe states corresponding to alignment with the inner binary (i.e., transition region (ap ≈ rL) is stable, we expect the planet’sorbitto ^ ^ → Lp,eq Lin) at small ap, and alignment with the outer companion evolve smoothly following the Laplace surface (i.e., regime a ^ ^ → > (i.e., Lp,eq Lout) at large ap. The transition between these two regime b regime c). For iin‐out 69°, however, the planet will orientations happens rapidly at the so-called “Laplace radius” encounter an instability when ap ≈ rL, and may undergo erratic evo- rL, obtained by setting Ωp,out = Ωp,in, and is given by lution, which may result in the planet being destroyed or ejected. In the LK+tide scenario for the formation of compact binaries, = μ 1 5 the final inner binary separation a depends on the properties of r = in a2 a3 [3] in,f L in out . ‐ 2Mout the outer companion (Mout, aout, and the initial inclination iin out) as well as on the short-range force effects between the inner binary Thus, there are three regimes (see Supporting Information for members (15). Thus, for a given stellar triple configuration, the a schematic depiction) for the planet’s equilibrium orienta- inner binary may or may not reach down to ain,L, depending on the > tion: (regime a) Ωp‐in Ωp‐out or binary-dominated regime value of ap (Fig. 1). If ain,f ain,L, or equivalently, if (a r ); (regime b) Ω ‐ ≈ Ω ‐ or transition regime (a r ); p L p out p in p L − =     Ω Ω 1 5 3=5 2=5 and (regime c) p‐out p‐in or companion-dominated regime < Mout aout ain,f [6] (a r ). ap 1.26 μ AU, p L ^ 4 in 30AU 0.03AU In general, however, the vector Lin is not fixed in space but L^ slowly precesses around out, owing to the torque from the outer the planet will never cross the intermediate regime (a ≈ r ), and L^ L^ p L companion (strictly speaking, both in and out precess around the it will thus remain “safe” (stable), regardless of the inclination total angular momentum vector of the system; however, for the i ‐ , surviving the orbital decay of its host binary. hierarchical configurations presented here, the outer orbit contains in out L^ most of the angular momentum of the system, implying that out is Evolution of Planetary Orbits Around Binaries Undergoing approximately fixed in space). This means that the plane normal to Lidov–Kozai Cycles with Tidal Friction L^ × L^ L^ in out, where the equilibrium orientation vector p,eq lives, is The greatest caveat to the application of classical Laplace equi- slowly rotating (see Supporting Information). This rotation rate is of librium is that the inner binary does not remain circular during order 3 Mout ain Ωin‐out ≡ nin , [4] Min aout qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = G = 3 where nin Min ain is the mean motion of the inner binary. 3=2 ASTRONOMY Note that Ωp‐out=Ωin‐out = ðap=ainÞ 1 in the companion- 3=2 dominated regime, and Ω ‐ =Ω ‐ = ðΩ ‐ =Ω ‐ Þða =a Þ = = p in in out p in p out p in ðr =a Þ5ða =a Þ3 2 1 in the binary-dominated regime. This L p p in ^ ^ means that the precession of Lin is always slow enough for Lp to adiabatically follow. In other words, the classical Laplace equilibrium formalism remains valid in the frame corotating ^ ^ ^ ^ with Lin, and the three vectors Lin, Lout,andLp,eq remain co- planar at all times. Because the evolution is adiabatic, if L^ starts ^ ^ p parallel to Lp,eq, it will remain parallel to the evolving Lp,eq at later times, provided that this equilibrium orientation is a stable solution (17). As studied by ref. 17, when iin‐out > 69°, circular orbits on the Laplace surface are unstable to linear perturbations in the planet’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 that above this critical value of iin‐out, planets cannot be placed at ap ≈ rL, because the resulting high eccentricities could bring them too close to the binary, at which point they may collide with the central 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- Ω Ω Ω sume that the binary remains circular during this process, and Fig. 1. The three relevant precession frequencies ( p‐in, p‐out,and in‐out)asa function of the shrinking binary semimajor axis a . The binary starts at semi- that the angle iin‐out remains unchanged. Because orbital decay in major axis ain,0 = 0.3 AU and circularizes at ain,f = 0.024 AU (vertical black line). takes place over a time scale tdecay much longer than the other The other parameters are M = M = 1M⊙, μ = 0.25, a = 30 AU, and relevant time scales (1=Ω ‐ ,1=Ω ‐ and 1=Ω ‐ ), the system in out in out p in p out in out e = 0. (Top)Thecasewitha = 2AU,and(Bottom)thecasewitha = 1 AU. In will evolve adiabatically. Thus, if the planet initially resides in the out p p Top, ain crosses ain,L = 0.097 AU (thick vertical gray line) during orbital decay, binary-dominated regime (Ωp‐in Ωp‐out, ap rL), and lives on L^ L^ then the planet transitions from the binary-dominated regime into the com- the Laplace surface ( p p,eq), it will transition to the compan- panion-dominated regime. In Bottom, a > a = 0.017 AU, and the planet Ω Ω in,f in,L ion-dominated regime ( p‐in p‐out, ap rL) through the in- will stay in the binary-dominated regime throughout the binary orbital decay. Ω ≈ Ω ’ termediate stage ( p‐in p‐out), as the inner binary s semimajor Note that in this example, ap = 1 AU is very close to the initial binary, and axis ain decreases. For a given value of ap, the transition occurs dynamical instabilities (not captured by secular calculations) might make the when ain passes through a critical (Laplace) value, survival of these planets difficult during the early Lidov−Kozai cycles of the binary.

Muñoz and Lai PNAS | July 28, 2015 | vol. 112 | no. 30 | 9265 Downloaded by guest on September 29, 2021 initialized on a at a = 1.5 AU with L^ aligned with ^ p p Lin. The parameters for the inner binary and the planet are chosen to roughly correspond to the discovered Kepler systems. The pa- rameters for the outer orbit are chosen to ensure that LK cycles are not suppressed by short-range forces and to guarantee the efficient orbital decay of the inner binary (15). In our calculations, the octupole term in the potential has been ignored in the evolution equations of the planet and the inner binary, a justified simplifi- cation because m0 = m1 and eout = 0. The inner binary experiences LK oscillations and circularizes within a Hubble time provided that enough tidal dissipation is present in the stars. The final (circu- larization) semimajor axis is ain,f = 0.053 AU (corresponding to an orbital period of 4.5 d). In this example, the planet initially resides in the binary-dominated regime, with Ωp‐in=Ωp‐out ≈ −5 65ðap=AUÞ ≈ 8.6, and ap=rL,0 ≈ 0.65. After the inner binary has circularized, 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 planet remains on a circular orbit throughout its entire evolution, despite the large variations in ein during the LK cycles. The longitude of nodes of the planet (not shown in the figure) closely follows that of the inner binary during the early LK cycles and after circulariza- tion, implying that, for a large fraction of the time, L^ is coplanar ^ ^ p with Lin and Lout. The planet’s inclination ip‐out also follows the binary inclination iin‐out during the early stage of the LK cycles (Fig. 2, Bottom Middle), but it decouples from the inner binary after ain has 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 a steady-state value. This final value, ip‐out ’ 14°, agrees with the equilibrium value of the end-state Laplace surface (with = = = Fig. 2. An example of the coupled evolution of an inner binary within a ain,f 0.053 AU and iin‐out,f 46°) evaluated at ap 1.5 AU. stellar triple (m0 = m1 = 0.5M⊙, Mout = 1M⊙, aout = 18 AU, and initial ain,0 = 0.3 We have carried out calculations for a range of values of ap for AU and iin‐out,0 = 73°) plus a planet with semimajor axis ap = 1.5 AU. The the same stellar triple configuration of Fig. 2. The results of different panels show (Top) binary semimajor axis ain (green) and planet these calculations are summarized in Fig. 3, which shows the semimajor 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); (Bottom Middle) inclination of the binary iin‐out (green) and inclination of the planet ip‐out (blue) with respect to the outer companion; and (Bottom) mu- tual inclination between the planet and the binary ip‐in. The eccentricity and inclination of the binary exhibit LK cycles for about 10% of the integration time, until short-range forces arrest these oscillations (freezing ein at high values), at which point a slow phase of orbital decay takes place. The planet starts in the binary-dominated regime (ap=rL,0 = 0.65). Its inclination ip‐out follows closely that of the inner binary iin‐out until rL crosses ap (note that the definition of rL in Eq. 3 does not take into account the eccentricity of the inner binary ein), at which point these two inclination angles decouple from each other. 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 a constant value of ∼ 32°.

orbital decay. Indeed, in the LK+tide mechanism (14, 15), the inner binary exhibits large oscillations in inclination and eccen- tricity under the influence of the external stellar companion. Thus, ^ ^ the binary axis Lin not only precesses around Lout but also un- dergoes nutation. The variation of the inner binary’s eccentricity = vector ein also affects the torque on the circumbinary planet. Fig. 3. Classical Laplace equilibrium surface (valid for ein 0) at the begin- To track the evolution of the planet’s orbit during the LK ning (red curve) and after circularization of the inner binary (orange curve) = = = = oscillations and orbital decay of the inner binary, we solve nu- for the triple configuration of Fig. 2 (m0 m1 0.5M⊙, Mout 1M⊙, ain,0 0.3 = = = = merically the secular equations of the planet’s eccentricity vector AU, 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- ep and angular momentum vector axis Lp (Supporting Informa- librium surface is unstable (17). The final Laplace surface is stable for all ap tion), along with the evolution equations of the stellar triple (the < secular equations of motion govern the evolution of the orbital because ip‐out,f 60°. The vertical dash-dotted lines indicate the Laplace radii elements instead of the position and velocity of individual at the beginning (rL,0) and end (rL,f ) of the binary orbital evolution. For the different values of ap, vertical arrows connect the initial and final states, bodies). We use the formalism of ref. 20 to follow the inner ’ ’ representing the evolution of the planet s inclination obtained from the binary s orbit and parametrize the stellar tidal dissipation rate numerical calculations. In each case, the planet orientation is initially aligned using the weak friction model with constant tidal lag time. In the with the local Laplace surface (or approximately aligned with the binary for following, we focus on a few representative examples and discuss ap K 1.5 AU); after the inner binary has decayed and circularized, the planet the general behavior for the evolution of the four-body system. inclination settles into a value coincident with final Laplace surface. Note Fig. 2 depicts a system where the stellar triple has parameters that, for illustrative purposes, we include values of a down to 1 AU; how- = = = = = p m0 m1 0.5M⊙, Mout 1M⊙, aout 18 AU, and eout 0andini- ever, dynamical stability dictates that only planets outside ap ≈ 4ain ≈ 1.2 AU tial values ain,0 = 0.3 AU and iin‐out,0 = 73°, and where the planet is [when ein ≈ 1 (19),] should survive the early LK cycles of the inner binary.

9266 | www.pnas.org/cgi/doi/10.1073/pnas.1505671112 Muñoz and Lai Downloaded by guest on September 29, 2021 ∘ 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 and aout = 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 different planetary 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. ASTRONOMY

Laplace surfaces at the beginning and at the end of the evolu- ap = 1.2 AU (Fig. 4, Left), the inclination angle ip‐out does not tion, when the inner binary is circular. In each case, the planet is evolve smoothly as the inner binary decays but suffers a jump as ^ initially aligned with the equilibrium orientation Lp,eq, which is in rL crosses ap, subsequently oscillating around a reference angle. near alignment with the inner binary for ap K 1.5 AU. We find Moreover, the rapidly grows until it starts that the planet’s inclination evolves smoothly for all these cases oscillating around a mean value of hepi ≈ 0.16, maintaining from as the binary experiences LK oscillations and orbital decay. then on a steady-state behavior. For a planet at ap = 1.8 AU (Fig. Despite the complexity of the “intermediate” states, in which the 4, Right), the orbital evolution is even more complex. In this case, binary develops large eccentricities and the standard Laplace the exponential growth in eccentricity does not saturate at a equilibrium is not well defined, we find that in the end, the moderate value. Instead, ep reaches values close to 1. The erratic planet’s inclination always lands on the final Laplace surface. evolution in ep is accompanied by a similar behavior in the Thus, these planets survive the orbital decay of the inner binary planet’s inclination ip‐out. Instead of oscillating around a mean but become inclined with respect to it by an angle given by (equilibrium) value, ip‐out covers the entire range ð0°, 180°Þ. The ip‐in,f = iin‐out,f − ip‐out,f (with iin‐out,f ≈ 46° for the parameters adopted high planet eccentricities reached in this case make it very un- in Figs. 2 and 3), where ip‐out,f matches the equilibrium inclination likely for the planet to survive the orbital decay of the inner of the final Laplace surface. Because the Laplace equilibrium binary. Such high eccentricities will inevitably bring the planet inclination angle decreases with increasing ap, we predict that the too close to the inner binary, a region that is known to be un- angle ip‐in of the planets that survive will increase monotonically stable (19, 21). In this case, ejections from the system or physical with increasing ap. collisions with the central stars are to be expected. As noted before, when the mutual inclination iin‐out between In Fig. 5, we show the initial and final inclinations (Fig. 5, the inner circular binary and the external companion is greater Top), and the respective final eccentricities (Fig. 5, Bottom), than 69°, a portion of the Laplace surface is unstable (17). In computed for a set of values of ap using the same stellar triple principle, a circumbinary planet may suffer a similar instability as configuration of Fig. 4. As in Fig. 3, these results are shown a binary with large initial iin‐out undergoes LK+tide orbital decay. together with the Laplace equilibrium surface solutions for the In Fig. 4, we show two examples (ap = 1.2 AU and ap = 1.8 AU for initial state ðain,0 = 0.3 AU and iin‐out,0 = 83°Þ and the final state Fig. 4, Left and Fig. 4, Right, respectively) of planets within a ðain,f = 0.026 AU and iin‐out,f = 65.3°Þ. Unlike Fig. 3, we find that, stellar triple with aout = 30 AU, ain,0 = 0.3 AU, and iin‐out,0 = 83° depending on ap, planet orbits do not always stay circular, and (the other parameters are the same as in Fig. 2). At this initial their inclinations ip‐out do not always land exactly on the final inclination, the inner binary attains very high eccentricities and Laplace surface. For ap K rL,f , planets end up very close (on can circularize very efficiently (alternatively, it requires rela- average) to the final Laplace surface (while exhibiting some tively small tidal dissipation in the stars to circularize within minor oscillations around it), and maintain a negligible eccen- −2 a Hubble time). The final binary separation is ain,f = 2.55 × 10 tricity. For ap J rL,f , planets suffer a small kick in eccentricity AU (period of 1.5 d), and the inclination angle freezes out as they cross the “transition” regime ðap = rLÞ, and their inclinat- at iin‐out,f = 65.3°. The behavior of the planets is markedly dif- ions oscillate with significant amplitude around a mean value that ferent from the one depicted in Fig. 2. For a planet located at is close, but not necessarily equal to, the one given by Laplace

Muñoz and Lai PNAS | July 28, 2015 | vol. 112 | no. 30 | 9267 Downloaded by guest on September 29, 2021 23), imposing an upper limit on the maximum eccentricity of the binary. In Fig. 6, we show the maximum eccentricity achieved by the inner binary as a function of planet mass obtained from integrations of the four-body secular system (see Supporting Information). For the example depicted in Fig. 2 (ap = 1.5 AU and mp,crit ’ 0.6MJ), we find that mp J 1MJ ’ 1.7mp,crit is enough to substantially suppress the oscillations in ein.For mp K 0.3MJ ’ 0.5mp,crit (about the mass of Saturn), the minimum pericenter separation of the binary ain,0ð1 − ein,maxÞ ≈ 0.09ain,0 is only 17% larger than 0.077ain,0, the value corresponding to mp = 0. Such a planet (mp K 0.3MJ) will only delay the orbital shrinkage of the inner binary, but not prevent it (see Supporting Information for an example). Throughout this paper, we have included only the quadrupole potential from the tertiary companion acting on the inner binary and 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 introduce more complex dynamical behaviors for the inner binary and for the planet (see, e.g., refs. 23–26). For example, in N-body cal- culations (which include high-order terms automatically), the planet may attain a nonzero eccentricity as the inner binary decays even in the moderate inclination case (see Supporting Information for one such example). A systematic study of these complex “high-order” effects is beyond the scope of this paper and will be the subject of future work. Discussion Fig. 5. (Top) Similar to Fig. 3, but for the triple configuration shown in Fig. 4 We have explored the orbital evolution of planets around bi- = = = = (ain,0 0.3 AU, iin‐out,0 83°, ain,f 0.026 AU, iin‐out,f 65.3°). (Bottom) The naries undergoing orbital decay via the LK+Tide mechanism corresponding final eccentricity of each planet. Error bars specify the oscil- J driven by distant tertiary companions. We have shown that lation amplitude around the mean value. Wide blue bands (for ap 1.5 AU) planets may survive the orbital decay of the binary for tertiary denote planet orbits with erratic behavior in inclination, covering the entire ‐ K range ½0°, 180°Þ, at any point during their evolution. Note that, for some companions at moderate initial inclinations (iin out,0 75°). In such cases, planets on circular orbits adiabatically follow an values of ap, planets do reach regular values in eccentricity and inclination even after having experienced erratic evolution during a finite period before equilibrium solution as the triple system evolves, becoming binary circularization; such cases are still depicted by blue bands, because misaligned with their host binary; the final misalignment angle their survival is deemed unlikely (see Supporting Information). ip‐in is a monotonically increasing function of the binary-planet

equilibrium (see Fig. 4, Left). At even larger apð J 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 ’ 0-1andIp‐out ’ 0°‐80°). Indeed, for large 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 and inclination) may last indefinitely or may end before circulari- zation of the inner binary has completed, in which case planets can exit the erratic phase at a random inclination (including angles > 90°). In either case, these planets, having experienced erratic, large-amplitude variations of ep, are likely to be ejected from 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 affect the dynamics of the inner binary to the point of suppressing the eccentricity oscillations caused by the tertiary (22). The planet-induced precession frequency of the binary is of order 3 Ωin‐p ’ ninðmp=MinÞ ðain=apÞ . The condition Ωin‐p ’ Ωin‐out al- lows us to define the critical planet mass     3 −3 ’ Mout ap aout [7] Fig. 6. (Top) Maximum eccentricity ein,max of the inner binary in the triple mp,crit 0.13MJ configuration of Fig. 3 achieved during the LK cycles as a function of planet 1M⊙ 1.5AU 30AU mass mp for three different values of the planet semimajor axis ap:1.2AU (blue), 1.5 AU (red), and 1.8 AU (orange). Vertical lines denote the value of (where M is the mass of Jupiter) above which the precession of J mp,crit (Eq. 7) for each of the different values of ap.(Bottom) Same as Top, the binary due to the planet is faster that due to the tertiary star. but for a triple configuration as in Fig. 5. In general, LK oscillations are en- J < 1 For small mp, the effects of a finite planet mass on the LK cycles tirely suppressed for mp 2mp,crit. For smaller planet mass (mp 2 mp,crit), the are qualitatively similar to those of other short-range forces (15, eccentricity oscillation amplitude is only slightly modified.

9268 | www.pnas.org/cgi/doi/10.1073/pnas.1505671112 Muñoz and Lai Downloaded by guest on September 29, 2021 distance ap. At higher inclinations ðiin‐out,0 J 80°Þ, the adiabatic and the outer companion, introducing additional complications to evolution is broken when planets encounter an unstable equi- the formation of planetary cores (30, 31). librium. Then the planet orbit can develop erratic behavior in As noted before, currently, no planets have been detected eccentricity and inclination. Very eccentric circumbinary orbits around eclipsing compact (Pin K 5 d) stellar binaries. Our work may be disrupted by the inner binary via dynamical instabilities, suggests that if planets are able to form within (moderately) resulting in either the ejection of the planet or its collision onto compact triples, they are likely to survive the tidal shrinkage of the stars. Interestingly, even in this high-inclination regime, we the central binary, evolving into inclined orbits. The detection of have found that some planets may evolve into stable, misaligned these misaligned circumbinary planets may be challenging. The and eccentric orbits. planets that survive the orbital decay of the binary lie close to/on In our scenario, the abundance of misaligned planets around the Laplace surface, which follows with the precession of the ^ ^ compact binaries depends on the frequency of moderate initial inner binary axis Lin with respect to the outer binary axis Lout. inclination stellar triples relative to those with high inclinations. The coupled precession of the inner binary and the planet or- High-inclination stellar triples may be the progenitors of the bits will produce short-lived “transiting windows,” but these win- majority of compact binaries, because the very high eccentricities dows appear periodically over very long time scales [of order reached by the inner binary make orbital decay faster. Our cal- 3 5 6 1=Ωin‐out ≈ Pinðaout=ainÞ ≈ 10 − 10 y]. An alternative detection culations suggest that planets within such high-inclination triples strategy is to look for eclipse timing variations. The have less chances of survival during the inner binary’s orbital on the inner binary exerted by a planet of mass mp introduces a decay. The efficiency of tidal decay depends on the dissipation timing signature (on the time scale of the planet’s orbital period) = time scale tV within the stars (see Supporting Information). We of magnitude ∼ ð3=8πÞP ðm =M Þða =a Þ3 2 (32, 33). For val- have found that dissipation time scales of order 20 − 50 y can in p in in p ues of mp ≈ 0.5MJ ≈ 0.0005Min, ap ≈ 1.0 AU, ain ≈ 0.05 AU, and circularize inner binaries with iin‐out,0 J 78° within a Hubble time, Pin ≈ 5 d, the maximum eclipse timing variation is of order ∼ 0.3 s, i ‐ ∼ t ’ − but if in out,0 70°, then V 1 5 y is required. However, given approaching the noise level for some nearby binaries but, in the large parameter space in orbital configurations, and the general, still below the detection limits for most eclipse timing uncertainty in realistic values of t (which may vary during stellar V detections (34). However, short cadence data with current ob- evolution), we cannot discard the possibility that some binaries, perhaps still undergoing orbital decay and circularization, may be servational capabilities might provide enough timing precision to part of moderate-inclination stellar triples, and may therefore be accomplish such measurements. candidate hosts to highly misaligned planets. An additional caveat to the abundance of misaligned circum- Note. During of the revision of our manuscript, we became aware binary planets that is not addressed in this work concerns the of a preprint by D. Martin, T. Mazeh, and D. Fabrycky, which likelihood of planets forming within inclined hierarchical triples addresses a similar issue (i.e., the dearth of planets around com- pact binaries) as our paper (35). with 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, planets ACKNOWLEDGMENTS. We thank Sarah Ballard, Konstantin Batygin, Matthew Holman, and Bin Liu for discussions and comments. We also thank the referee, would be confined to form between 1 AU and 10 AU. In addition Daniel Fabrycky, for valuable comments and suggestions. This work has been to disk truncation and warping (29), planetesimal dynamics in this supported in part by National Science Foundation Grant AST-1211061 and

systems could be affected by the tidal forcing of the inner binary NASA Grants NNX14AG94G and NNX14AP31G. ASTRONOMY

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