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Survival of Planets Around Shrinking Stellar Binaries Survival of planets 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 Planet Inside a Stellar Triple suggests that planets can form efficiently around binary stars. 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 orbit each other with a semimajor axis aout ain. compact binaries are believed to have evolved from wider orbits 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-mass ratio 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 star 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 orbital period 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 26, 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.
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