# Survival of planets around shrinking stellar binaries

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

## Significance

The detection of planets around binary stars (sometimes called “Tatooine planets”) in the last few years signified a major discovery in astronomy and posed a significant challenge to our understanding of planet formation. So far, the discovered circumbinary planets orbit relatively wide stellar binaries (with binary orbital period greater than 7 d) and have their orbital axes aligned with the binary axes. The theoretical/numerical work reported in this paper suggests that there may be a new population of circumbinary planets, which orbit around more-compact binaries (with periods less than a few days) and have their orbital axes misaligned with the binary axes. Current observational strategy inevitably misses this population of Tatooine planets, but future observations may reveal their existence.

## Abstract

The discovery of transiting circumbinary planets by the Kepler mission suggests that planets can form efficiently around binary stars. None of the stellar binaries currently known to host planets has a period shorter than 7 d, despite the large number of eclipsing binaries found in the Kepler target list with periods shorter than a few days. These compact binaries are believed to have evolved from wider orbits into their current configurations via the so-called Lidov–Kozai migration mechanism, in which gravitational perturbations from a distant tertiary companion induce large-amplitude eccentricity oscillations in the binary, followed by orbital decay and circularization due to tidal dissipation in the stars. Here we explore the orbital evolution of planets around binaries undergoing orbital decay by this mechanism. We show that planets may survive and become misaligned from their host binary, or may develop erratic behavior in eccentricity, resulting in their consumption by the stars or ejection from the system as the binary decays. Our results suggest that circumbinary planets around compact binaries could still exist, and we offer predictions as to what their orbital configurations should be like.

To date, the *Kepler* spacecraft has discovered eight binary star systems harboring 10 transiting circumbinary planets (1⇓⇓⇓⇓⇓⇓–8). These systems have binary periods ranging from 7.5 d to *Kepler* mission. However, the shortest-period binary hosting a planet is Kepler-47(AB), with 7.44 d, despite the fact that nearly 50*Kepler* data have periods shorter than 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 (

In synthetic population studies (15), stellar binaries with periods shorter than *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 planet until binary circularization is reached and binary separation is shrunk by an order of magnitude. We show that the tertiary companion can play a major role in misaligning and/or destabilizing the planet as the binary shrinks.

## A Planet Inside a Stellar Triple

Consider a planet orbiting a circular stellar binary of total mass *Supporting Information* for a schematic depiction) for the planet’s equilibrium orientation: (regime a)

In general, however, the vector *Supporting Information*). This rotation rate is of order

As studied by ref. 17, when

Now consider what will happen to the planet’s orbit as the inner binary undergoes orbital decay. For simplicity, let us assume that the binary remains circular during this process, and that the angle **3** and solving for

In the LK+tide scenario for the formation of compact binaries, the final inner binary separation

## Evolution of Planetary Orbits Around Binaries Undergoing Lidov–Kozai Cycles with Tidal Friction

The greatest caveat to the application of classical Laplace equilibrium is that the inner binary does not remain circular during orbital decay. Indeed, in the LK+tide mechanism (14, 15), the inner binary exhibits large oscillations in inclination and eccentricity under the influence of the external stellar companion. Thus, the binary axis

To track the evolution of the planet’s orbit during the LK oscillations and orbital decay of the inner binary, we solve numerically the secular equations of the planet’s eccentricity vector *Supporting Information*), along with the evolution equations of the stellar triple (the secular equations of motion govern the evolution of the orbital elements instead of the position and velocity of individual bodies). We use the formalism of ref. 20 to follow the inner binary’s orbit and parametrize the stellar tidal dissipation rate using the weak friction model with constant tidal lag time. In the following, we focus on a few representative examples and discuss the general behavior for the evolution of the four-body system.

Fig. 2 depicts a system where the stellar triple has parameters *Kepler* systems. The parameters 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 simplification because *Bottom Middle*), but it decouples from the inner binary after *Bottom*) and the planet inclination has settled onto a steady-state value. This final value,

We have carried out calculations for a range of values of

As noted before, when the mutual inclination *Left* and Fig. 4, *Right*, respectively) of planets within a stellar triple with *Left*), the inclination angle *Right*), the orbital evolution is even more complex. In this case, the exponential growth in eccentricity does not saturate at a moderate value. Instead,

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 *Left*). At even larger *Right*): Both *Supporting Information*). Erratic evolution (in eccentricity and inclination) may last indefinitely or may end before circularization of the inner binary has completed, in which case planets can exit the erratic phase at a random inclination (including angles

In the above, our calculations have ignored the mass of the circumbinary planet *Supporting Information*). For the example depicted in Fig. 2 (*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 companion has zero orbital eccentricity. For general companion eccentricities, 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 calculations (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

We have explored the orbital evolution of planets around binaries undergoing orbital decay via the LK+Tide mechanism driven by distant tertiary companions. We have shown that planets may survive the orbital decay of the binary for tertiary companions at moderate initial inclinations (

In our scenario, the abundance of misaligned planets around compact binaries depends on the frequency of moderate initial inclination stellar triples relative to those with high inclinations. High-inclination stellar triples may be the progenitors of the majority of compact binaries, because the very high eccentricities reached by the inner binary make orbital decay faster. Our calculations suggest that planets within such high-inclination triples have less chances of survival during the inner binary’s orbital decay. The efficiency of tidal decay depends on the dissipation time scale *Supporting Information*). We have found that dissipation time scales of order

An additional caveat to the abundance of misaligned circumbinary planets that is not addressed in this work concerns the likelihood of planets forming within inclined hierarchical triples with

As noted before, currently, no planets have been detected around eclipsing compact (

### Note.

During of the revision of our manuscript, we became aware of a preprint by D. Martin, T. Mazeh, and D. Fabrycky, which addresses a similar issue (i.e., the dearth of planets around compact binaries) as our paper (35).

## Equations of Motion

Consider a binary (total mass

and

In our actual calculations, we will set the outer orbit’s eccentricity

From the potentials

and**1** and **2** in *A Planet Inside a Stellar Triple*,

By setting **S3a**, and setting **3**), because

The three equilibrium regimes for the *A Planet Inside a Stellar Triple* correspond to (regime a) **S6**, when

In our numerical calculations, we directly integrate Eqs. **S3a** and **S3b** together with the evolution equations of the hierarchical triple as the inner binary undergoes LK oscillations with tidal dissipation. In principle, this system consists of 24 coupled differential equations (involving the vectors

For the evolution of the vectors *Q* scales proportionally with *k* (the classical apsidal motion constant) is set to 0.014. We see that one to two orders of magnitude of variation in *Q* values for different types of stars. The assumption of pseudosynchronization and spin-orbit alignment also introduces uncertainties, although the dominant uncertainty still lies in the value of *Discussion*).

## Erratic Evolution and Chaotic Behavior

In Fig. S3, we show two integrations for planets within the stellar triple configuration of Fig. 5. Fig. S3, *Top* shows the eccentricity and inclination evolution of a planet with *Bottom* shows the same for a planet with

## Effect of Finite Planet Mass

In the calculations above, the mass of the planet **S7**. These extra terms have the same functional form as the terms arising from the tidal potential of the tertiary

## Example of an *N*-Body Integration

Although long-term direct *N*-body integrations are computationally costly, one can combine them with the output of the secular solutions to study the detailed behavior of the planetary orbit as the binary decays. Fig. S5 shows the *N*-body result of a dissipative binary performed using the Mercury code (36). We have added short-range forces following ref. 37, including stellar tides, general relativity, and tidal dissipation. This integration is started once the binary has undergone appreciable orbital decay, but before the planet has crossed the Laplace radius

As seen from Fig. S5, before crossing the Laplace radius, the planet’s inclination adiabatically follows that of the inner binary while maintaining zero eccentricity, in full agreement with our secular results. After

## Acknowledgments

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

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: dmunoz{at}astro.cornell.edu.

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. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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

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