Birth environment of circumbinary planets: are there circumbinary planets on inclined orbits?

ABSTRACT

The distribution of the orbital inclination angles of circumbinary planets (CBPs) is an important scientific issue, and it is of great significance for estimating the occurrence rate of CBPs and studying their formation and evolution. Although the CBPs currently discovered by the transit method are nearly coplanar, the true distribution of the inclinations of CBPs is still unknown. Previous research on CBPs has mostly regarded them as isolated binary-planet systems, without considering the birth environment of their host binaries. It is generally believed that almost all stars are born in clusters. Therefore, it is necessary to consider the effects of the close encounters of stars on CBP systems. In this paper, we discuss how the close encounters of fly-by stars affect the inclinations of CBPs. Based on extensive numerical simulations, we have found that CBPs in a close binary with a spacing of ∼0.2 au are almost unaffected by fly-by stars. Their orbits remain coplanar. However, when the spacing of the binary stars is greater than 1 au, two to three fly-bys of an intruding star can excite a considerable inclination, even for a CBP near the unstable boundary of the binary. For CBPs in the outer region, the fly-by of a single star can excite an inclination to more than 5°. In particular, CBPs in near polar or retrograde orbits can naturally form through binary–star encounters. If close binaries are born in open clusters, our simulations suggest that there may be high-inclination CBPs in binaries with a spacing >1 au.

celestial mechanics, binaries: general, planetary systems

1 INTRODUCTION

Circumbinary planets (CBPs) are one of the miraculous discoveries of exoplanet exploration. At the time of writing, more than 20 CBPs have been found (Schwarz et al. 2016).¹ The most influential subset of these contains the 11 planets detected by Kepler, which constitute a small research sample. One remarkable feature of the orbits of Kepler’s CBPs is coplanarity (i.e. the planes and their host binary are nearly in the same plane). The inclination angle I between the two orbits is less than 2|${^{\circ}_{.}}$|5 (Kostov et al. 2014). This feature is partly a result of the observational selection effect, because planets orbiting in the binary plane can be more easily detected by the transit method. A very interesting question is whether there are CBPs on inclined orbits hidden in the Universe. The distribution of the orbital inclination angles of CBPs is an important scientific issue. For example, occurrence rates of CBPs depend critically on the inclination distribution (Armstrong et al. 2014). If the underlying planetary inclination distribution is isotropic, then the occurrence rate of CBPs is significantly greater than the analogous rate for single stars. Martin & Triaud (2014) have investigated the possibility of finding planets that are transiting non-eclipsing binaries. They have found that there are potentially many of these hidden in the Kepler photometric data. Zhang & Fabrycky (2019) have provided a new tool for discovering potential polar CBPs, or misaligned CBPs of milder inclinations, from the existing Kepler data set of eclipse timing variations. Perhaps CBPs on inclined orbits will be detected in the near future.

Although no CBPs have been found on high-inclination orbits (I > 5°), some circumbinary discs (CBDs) have been discovered on high-inclination orbits in recent years. For example, the CBD in the IRS 43 system has an inclination I > 40° (Brinch et al. 2016; Czekala et al. 2019). The CBD 99 Herculis and a planet-forming CBD in the young HD 98800 system are thought to have a polar configuration I ∼ 90° (Kennedy et al. 2012, 2019). Theoretical research shows that the evolution of a CBD tends to have two extreme cases (Brinch et al. 2016; Martin & Lubow 2017, 2018; Lubow & Martin 2018; Zanazzi & Lai 2018). If the initial inclination angle of the disc is small, the result of the evolution is that the disc tends to be coplanar with the binary star. If the initial inclination angle of the disc is large (i.e. >40°; Martin & Lubow 2017) and the eccentricity of the binary star is non-negligible, then the disc will evolve into a polar orbit. The presence of high-inclination discs indicates that high-inclination CBPs might exist. However, the above conclusions are based on the evolution of the isolated CBD systems themselves. Besides, they can only evolve to high-inclination orbits in specific initial configurations, such as a high initial inclination and a large eccentricity for the binary. Other mechanisms are still needed to explain why the CBD has a large inclination initially. It is worth noting that the semimajor axes of binaries with high-inclination discs found already are relatively large (several tens of au), while the discs around close binaries are nearly coplanar (Kennedy et al. 2012; Czekala et al. 2019). For the Kepler CBPs, the semimajor axes of their host binaries are small, about 0.1–0.2 au. One possibility is that the discs around well-spaced binaries are susceptible to the surrounding environment, such as th close encounters with fly-by stars.

It is generally believed that most stars, and therefore most planetary systems, were born in clusters or associations (Clarke, Bonnell & Hillenbrand 2000; Lada & Lada 2003; Pfalzner 2013; Hao, Kouwenhoven & Spurzem 2013; Cai et al. 2017). Some open clusters later dissolved, forming the current field stars. A notable example is that our own Solar system may have formed in an open cluster. The chemical composition of objects in the Solar system, along with the orbital elements of Sedna, suggest that the Sun formed in a cluster with roughly 10³–10⁴ stars (Adams 2010). Other work even suggests that the Sun was born in a massive cluster (10⁴–10⁵ stars) because a cluster of several thousand stars is much too small to produce a nearby massive supernova (Dukes & Krumholz 2012). Malmberg, Davies & Heggie (2011) considered the effect of a close encounter between stars on the planetary systems in a single star system. They found that the fly-by of a star is one of the possible mechanisms for explaining the generally high eccentricity of exoplanes (mainly discovered by radial velocity methods). It is interesting to ask how the cluster environment affects CBPs. In fact, the first CBP discovered, PSR B1620–26, is located in a globular cluster (Sigurdsson et al. 2003). According to the theory of star formation, it is impossible to form a close binary directly (Moe & Kratter 2018). One of the explanations for this is that they are formed by interaction with other stars (Martin, Mazeh & Fabrycky 2015; Hamers, Perets & Portegies Zwart 2016; Moe & Kratter 2018), which means that they were in a star-dense environment previously. A cluster provides such a possibility. If a CBP was born in an open cluster, how do the close encounters with stars affect their orbital configuration? In particular, how do the fly-bys of stars affect the inclination distribution of CBPs? These questions are the focus of this work.

2 MODEL AND METHOD

We consider how the fly-by of a star affects the inclination of a CBP. The CBP family found by Kepler is used as the reference to set the parameters of the binary star and the planet. The planet is initially on a coplanar and circular orbit. Its mass is 1 Saturn mass. The semimajor axes of the binaries in theKepler CBP sample are about a_B ∼ 0.2 au. We take 0.2 au as the lower limit of a_B. According to Trilling et al. (2007), the CBD around close binaries that have a semimajor axis of 3 au is ubiquitous. The upper limit of a_B is set as 3 au. The mass ratio of the binary m₂/m₁ is 0.5. Here, m₁ and m₂ are the mass of the primary and secondary star, respectively. The eccentricity of the binary is e_B = 0.3.

There is a stable boundary (a_c) around the binary beyond which the orbit of the planet is long-term stable (except for some unstable islands; Holman & Wiegert 1999):

$$\begin{eqnarray} a_{\rm c} &=& [{1.6 + 4.12\mu + 5.1{e_{\rm B}} - 4.27\mu {e_{\rm B}}}\\ && -2.22e_{\rm B}^2 - 5.09{\mu ^2} + 4.61{\mu ^2}e_{\rm B}^2] \,a_{\rm B}. \end{eqnarray}$$

(1)

Here, μ = m₂/(m₁ + m₂), anda_B and e_B are the semimajor axis and eccentricity of the binary, respectively. The semimajor axes of the CBPs found by Kepler are distributed between 1.1 a_c and 7.4 a_c. Actually, most of them cluster in the vicinity of the unstable boundary of the binary, ∼1.1 a_c (Gong & Ji 2017). It is generally believed that CBPs formed in the outer region of the CBD and then migrated to their current locations (Thun & Kley 2018). Kepler-1647b has the longest period in the still small family of CBPs (∼1100 d, a_p = 7.4 a_c; Kostov et al. 2016), and it might not have undergone significant migration. In this work, we explore a_p = 1.1, 4.1 and 7.1 a_c. Their values in au are shown in Table 1. The orbital phase angles of the planets and the intruding star are randomly and uniformly distributed between 0° and 360°. Like most Kepler CBP systems, we only consider a single-planet system.

Figure 4.

Same as Fig. 2. Data are from Table 3. The pericentre distance of the fly-by star is 100 < q < 1000 au.

Table 1.

The semimajor axis (in au) of the CBP that we consider. The results come from equation (1). The eccentricity e_B and mass ratio μ of binaries are 0.3 and one-third, respectively.

aB (au)	ap = 1.1 ac	ap = 4.1 ac	ap = 7.1 ac
0.2	0.74	2.8	4.8
1.0	3.7	13.8	24
3.0	11	41	71

We focus on stellar perturbers in parabolic orbits, which is common even for ONC-like clusters (Olczak, Pfalzner & Eckart 2010; Pfalzner 2013; Xiang-Gruess 2016). It should be noted that in more dense clusters most stellar fly-bys are hyperbolic (Spurzem et al. 2009; Cai et al. 2017). For simplicity, we do not consider the hyperbolic orbit here. The orbit of the stellar perturber is described by five parameters: the pericentre distance q, the inclination i, the longitude of ascending node Ω, the argument of pericentre ω and the true anomaly f (Murray & Dermott 1999). All the parameters are relative to the barycentre of the binary.

We use mercury_ras (Smullen, Kratter & Shannon 2016) for numerical integration. It is a modified version of mercury (Chambers 1999) that can be used to simulate a CBP system. The code has been well tested in our former work (Gong & Ji 2018). We added a fly-by star to the system. The parabolic orbits of fly-by stars had been checked before the simulations. We found that although there is a negligible pulse in the eccentricity e₃ of the fly-by star near the pericentre, the intruding star is still in the near parabolic orbit e₃ ≈ 1 after passing it. This means that the quadrupole moment of the binary has little effect on the orbit of the fly-by star in the cases we explored. Fig. 1 shows an example. In our model, the fly-by star flies over the binary-planet system at an initial distance of 10 000 au from the barycentre of the binary. When the star passes its pericentre and the distance between it and the binary exceeds 1000 au again, we record the orbital parameters of the planet.

Figure 1.

The eccentricity evolution of the fly-by star. Top panel: the time evolution of the eccentricity of the fly-by star. Middle panel: zoomed-in version of part of the top panel to show details. Bottom panel: the time evolution of the distance of the intruding star from the barycentre of the binary. The pericentre distance q of the fly-by star is about 60 au. The semimajor axis of the binary is a_B =3 au.

The long-term stability of the surviving planet is judged by the condition q_p = a_p(1 − e_p) > a_c (the periastron of the planet is larger than a_c). Although the value of a_c in Holman & Wiegert (1999) was derived from the coplanar configuration, it can still be applied to the non-coplanar configuration, as shown in Pilat-Lohinger, Funk & Dvorak (2003). They found that the inclination of the CBP has little effect on the stable boundary. Therefore, we still use the criterion in Holman & Wiegert (1999) to check the orbital stability of the surviving CBPs.

As done in Malmberg et al. (2011), we divided fly-bys into two different regimes, depending on q: the strong regime (q < 100 au) and the weak regime (100 < q < 1000 au). When q is very small, the binary may be disrupted. We discarded these cases when they occurred in the simulation. In other words, the criterion for us to judge whether a simulation is successful is that the close binary is still intact after star fly-bys.

The mass of the intruder star also has an effect on the simulation results. In most simulations, we take m₃ = 1 M_⊙. In principle, a wide variety of stars with masses ranging from 0.08 M_⊙ up to 100 M_⊙ might act as a fly-by star in a cluster (Pfalzner et al. 2018). For example, it is generally believed that the Solar system can closely encounter with a star with a mass of 25 M_⊙ (Adams 2010). However, the more massive the stars, the fewer they are in a cluster. We just take |$m_{3}= 20\, \mathrm{M}_{\odot }$| as a representative of a high-mass perturber. The inclination of the fly-by star (relative to the binary plan) is randomly distributed between 0° and 180°. All initial phase angles of the planets and the intruding star were assigned randomly and uniformly from 0 to 2π.

3 NUMERICAL RESULTS

3.1 A single fly-by

3.1.1 q < 100 au

A fly-by that is this close can occur in dense star clusters or in the inner part of an open cluster. Pfalzner et al. (2018) showed that a close fly-by of 50 < q < 100 au by a neighbouring star can reproduce the properties of the Solar system. Numerical simulations have shown that in typical open clusters in the solar neighbourhood, which contain hundreds or thousands of member stars, 10–20 per cent of stars with mass ≥1 M_⊙ witness a fly-by < 100 au (Malmberg et al. 2007; Li, Mustill & Davies 2019). In our simulation, q of the fly-by star is evenly and randomly distributed between 0 and 100 au, and the inclination angle is randomly between 0° and 180°. The percentage of planets with an inclination greater than 5° (hereafter PGT5) after a fly-by of a single star is shown in Table 2. Each pair of statistical values in Table 2 is based on 1000 realizations. The total number of realizations is 72 000 in this work.

Table 2.

The percentage of planets with an inclination greater than 5° after a fly-by of a single star. The semimajor axis of the binary star is 0.2, 1 and 3 au, and the eccentricity is 0.3. The masses are m₁ = 1 M_⊙ and m₂ = 0.5 M_⊙. The initial semimajor axis of the planet is taken as 1.1, 4.1 and 7.1 a_c, where a_c is the unstable boundary around the binary. The initial eccentricity of the planet is 0. We considered two types of fly-by stars, m₃ = 1 M_⊙ and m₃ = 20 M_⊙. At the beginning of the simulation, the planet and the binary are coplanar. The pericentre distance of the fly-by star is uniformly and randomly distributed between 0 and 100 au. The lower limit of q satisfies the fact that the intruding star does not destroy the binary system. The initial inclination of the fly-by star relative to the binary orbital plane is randomly distributed between 0° and 180°. Each pair of data values in the table represents the PGT5 and the maximum inclination of the surviving planets, and is a statistical result of 1000 realizations.

m3 (M⊙)	aB (au)	ap = 1.1 ac	ap = 4.1 ac	ap = 7.1 ac
1	0.2	0.3%, 18°	2.0%, 80°	3.6%, 75°
	1.0	1.1%, 49°	10.6%, 119°	21.5%, 159°
	3.0	2.7%, 35°	39%, 163°	46%, 173°
20	0.2	0.2%, 13°	2.1%, 52°	2.9%, 125°
	1.0	0.7%, 86°	13%, 164°	26.8%, 128°
	3.0	4.3%, 84°	32.7%, 175°	39.4%, 177°

To clearly show the trend of the PGT5, we plot the values in Fig. 2. It can be seen from Fig. 2 that the percentage of planets with an inclination greater than 5° (i.e. PGT5) is gradually increasing as a_B increases (or see the respective column in Table 2). The maximum inclination of CBPs is also gradually increasing, albeit with some statistical randomness. In the same binary system, the further away the planet is, the easier it is for the inclination to be excited. That is, the PGT5 gradually increases for a_p from 1.1 to 7.1 a_c. A single fly-by can produce planets on retrograde orbits (i_p > 90°). For example, in the case of a_B = 1 au, a_p = 4.1 a_c = 13.8 au and the maximum inclination angle is 119°. We also considered a massive intruder with m₃ = 20 M_⊙. On the whole, the PGT5 increases slightly. In Fig. 3, we give an example of orbital evolution.

Figure 2.

The fraction of planets with an inclination greater than 5° after a fly-by by a single star. Data are from Table 2. Red squares, green dots and blue triangles represent the data of a_B = 0.2 au, 1 au and 3 au, respectively. Points connected by solid lines represent the case of m₃ = 1 M_⊙. Points connected by dashed lines represent the case of m₃ = 20 M_⊙.

Figure 3.

Orbital evolution of a circumbinary planet. To show the details, we only simulate the time before and after the fly-by. The time evolution of the semimajor axis, eccentricity and inclination of the planet is shown in the top three panels. The orbit of the intruding star is shown in the bottom panel. The parameters of the binary are m₁ = M_⊙, m₂ = 0.5 M_⊙, a_B = 1 au, e_B = 0.3. The planet is initially on a circular and coplanar orbit. Its semimajor axis is a_p = 7.1 a_c ≈ 24 au. The intruding star is m₃ = 1 M_⊙ with the pericentre distance q < 100 au. After the fly-by, the planet is excited to a high-inclination orbit (i_p ≃ 60°). Its orbit satisfies q_p > a_c, where q_p = a_p(1 − e_p) = 9.3 au and a_c = 3.4 au.

Now we focus on the Kepler CBPs (i.e. a_B ∼ 0.2 au, a_p ∼ 1.1 a_c = 0.74 au). The PGT5 is 0.3 per cent for an intruder with m₃ = 1 M_⊙. Even at the location of 7.1 a_c = 4.8 au from the binary, the percentage is still less than 5 per cent. For a massive fly-by star m₃ = 20 M_⊙, the PGT5 does not change much. For Kepler-like CBPs, the ratio is 0.2 per cent. The PGT5 of planets at 7.1 a_c = 4.8 au is only 2.9 per cent. In fact, we also tested fly-by stars with masses of 30, 50 and 100 M_⊙, and the PGT5 is still <1 per cent. This is because the PGT5 mainly depends on the q of the fly-by star. Only a small enough q can cause a significant orbit change for the inner planets. For a fly-by star with a large mass, its q cannot be too small because the binary itself will disintegrate for this type of close encounter. In summary, our calculations have shown that for planets similar to those discovered by Kepler, even if they were born in an open cluster, their inclination is almost unaffected by fly-by stars. In other words, the surviving CBPs are nearly coplanar.

3.1.2 100 au <q < 1000 au

When the q of a fly-by star is greater than 100 au and less than 1000 au, a solar-mass intruder has little effect on the inclination of the CBP (see Table 3 and Fig. 4). Only planets with a_p ≥ 4.1 a_c = 41 au in the binaries of a_B = 3 au can be excited to an inclination of more than 5°. In other cases, the PGT5 is 0.

Table 3.

The percentage of planets with an inclination greater than 5° after a fly-by of a single star. The pericentre distance of the fly-by star is uniformly and randomly distributed between 100 and 1000 au. The other parameters are the same as in Table 2.

m3 (M⊙)	aB (au)	ap = 1.1 ac	ap = 4.1 ac	ap = 7.1 ac
1	0.2	0%, 0.005°	0%, 0.1°	0%, 0.25°
	1.0	0%, 0.15°	0%, 1.3°	0%, 3°
	3.0	0%, 0.68°	0.5%, 7.5°	4.0%, 35°
20	0.2	0%, 0.01°	0%, 0.35°	0%, 0.8°
	1.0	0%, 0.92°	2.3%, 7.1°	9.5%, 65°
	3.0	0.1%, 5.6°	18.1%, 59°	20.6%, 99°

For fly-by stars with m₃ = 20 M_⊙, the PGT5 increases compared with the m₃ = M_⊙ case. For the binaries of a_B ∼ 0.2 au, the PGT5 is still zero. The fly-by stars have little effect on the planetary systems. For a_B = 3 au, the orbit of the outer planet is easily excited to a large inclination. For example, for a_p = 4.1 a_c = 41 au, about one-fifth of the planets are excited to an inclination of more than 5°. For a_p = 7.1 a_c = 71 au, the PGT5 is |$\sim 21{{\ \rm per\ cent}}$| with a maximum inclination of 99°.

3.2 Multiple fly-bys

So far, we have only discussed a single fly-by. Multiple fly-bys can also occur in an open cluster. For example, Malmberg et al. (2011) showed the average number of fly-bys per Sun-like star is four in a cluster with an initial number of stars of 700 and an initial half-mass radius of 0.38 pc. Besides, the more encounters a star undergoes, the smaller its fraction will be. For example, more than 60 per cent of the stars that have undergone fly-bys in their reference cluster experience ≤5 fly-bys. Close binaries (a_B ≤ 3 au) that we consider in this work are known as ‘hard binaries’ in cluster dynamics (Malmberg et al. 2007). They are more tightly bound and will not easily be broken up when they encounter another star. So the encounter rate of a close binary with a single star would be similar to the star–star encounter rate mentioned above. We have explored the effect of two to three fly-bys on the orbits of CBPs in the present work. Certainly, during the dissolution of a long-lived cluster, many more encounters exist. On the one hand, as mentioned above, their occurrence rate decreases significantly as the encounter number increases. On the other hand, with the dissolution of a cluster, the average encounter distance becomes large. On average, despite the encounter number increasing, the effects of fly-by stars on planetary systems are weak. These issues complicate the problem and are beyond the scope of our work.

We performed the simulations of multiple fly-bys as follows. After the first fly-by, we recorded the final position and velocity of the binary stars and planet. The data of the intruding star are discarded because these will have little effect on the orbit of the planet (r₃ ≥ 1000 au). We started a new run. These data are used as the initial condition for binary and planet. In addition, we added a new random fly-by star in the system. As in the first fly-by, the new fly-by star flies over the binary-planet system at an initial distance of 10 000 au from the barycentre of the binary. The third fly-by was performed in a similar way.

For fly-by stars with q < 100 au, the results of the three fly-bys are shown in Table 4. F1, F2 and F3 represent the results of the first, second and third fly-by, respectively. For the case of a_B = 3.0 au, after three fly-bys, most planets in the outer region are scattered out of the system. The few surviving planets cannot give a meaningful statistical result, so we discard these data. The fractions of planets with an inclination greater than 5° after the first, second and third fly-by are given in Fig. 5. In general, as the number of fly-bys increases, the PGT5 becomes larger. The larger a_B and a_p are, the larger PGT5 is.

Figure 5.

The fraction of planets with an inclination greater than 5° after the first, second and third fly-by. Data are from Table 4. The squares, circles and triangles represent the results of a_B = 0.2, 1 and 3 au, respectively. The results of the first, second and third fly-by are shown in red, green and blue, respectively. The pericentre distance of the fly-by star is q < 100 au.

Table 4.

The result of multiple fly-bys. F1, F2 and F3 represent the results of the first, second and third fly-by, respectively. For multiple fly-bys, we only consider a fly-by star with a mass of 1 M_⊙. The pericentre distance of the fly-by star is q < 100 au, as in Table 2.

aB (au)	Fly-by times	ap = 1.1 ac	ap = 4.1 ac	ap = 7.1 ac
0.2	F1	0.3%, 18°	2%, 80°	3.6%, 75°
	F2	0.3%, 29°	6.1%, 160°	9.7%, 169°
	F3	0.6%, 20.1°	7.3%, 150°	16.4%, 116°
1.0	F1	1.1%, 49°	10.6%, 119°	21.5%, 159°
	F2	0.9%, 26°	23.6%, 131°	39.6%, 171°
	F3	1.4%, 37°	30.4%, 144°	57.2%, 169°
3.0	F1	2.7%, 35°	39%, 163°	46%, 173°
	F2	4.5%, 18.2°	36%, 165°	–
	F3	9.7%, 56°	–	–

For a planetary system similar to a Kepler CBP (i.e. a_B = 0.2 au, a_p = 1.1 a_c = 0.74 au in Table 4), the PGT5 is still less than 1 per cent. After three fly-bys, PGT5 = 0.6 per cent and the maximum inclination is 19°. Only for the planets in the outer region (a_p = 7.1 a_c = 4.8 au), are more than 10 per cent of the planets on orbits greater than 5° after three fly-bys.

Table 5 and Fig. 6 give the results of 100 < q < 1000 au. For intruding stars of 1 M_⊙, the effect of multiple fly-bys on the planetary system is still limited. For the binary of a_B = 0.2 and 1 au, the PGT5 is 0. Only for systems with a_B = 3 au and a_p = 7.1 a_c = 71 au, is a considerable PGT5 obtained after three fly-bys. For three successive fly-bys, the results of PGT5 are 4, 10.4 and 15.9 per cent, respectively. The resulting maximum inclination is 39°.

Figure 6.

Conventions are as in Fig. 5. Data are from Table 5. The pericentre distance of the fly-by star is 100 < q < 1000 au.

Table 5.

The result of multiple fly-bys. The pericentre distance of the fly-by star is uniformly and randomly distributed between 100 and 1000 au. Other parameters are the same as in Table 4.

aB (au)	Fly-by times	ap = 1.1 ac	ap = 4.1 ac	ap = 7.1 ac
0.2	F1	0%, 0.005°	0%, 0.105°	0%, 0.25°
	F2	0%, 0.004°	0%, 0.14°	0%, 0.31°
	F3	0%, 0.005°	0%, 0.15°	0%, 0.32°
1.0	F1	0%, 0.15°	0%, 1.32°	0%, 3°
	F2	0%, 0.18°	0%, 1.35°	0%, 3.3°
	F3	0%, 0.19°	0%, 1.31°	0%, 2.9°
3.0	F1	0%, 0.68°	0.5%, 7.5°	4%, 35°
	F2	0%, 0.74°	1%, 8.5°	10%, 36°
	F3	0%, 0.93°	2.2%, 7.7°	16%, 39°

4 SUMMARY

It is generally believed that most stars are born in a cluster. The nascent planetary system of these stars is vulnerable to the perturbations of other stars. We have considered the effects of star fly-by(s) on the inclination of CBPs. Our main conclusions are as follows.

CBP systems with a_B ∼ 0.2 au, similar to those discovered by Kepler, are almost unaffected by a fly-by of a single star. Their orbits remain coplanar. Even a relatively close fly-by, or a massive intruder, has little effect on the inclination of the planet. Our simulations have also shown that several successive fly-bys do not cause significant inclination change for CBPs in tight binaries. These results imply that the orbits of planets similar to those discovered by Kepler, even if they were born in an open cluster, remain nearly coplanar.

However, for binaries with a_B > 1 au, the planets in the outer region (a_p > 4.1a_c) will be affected by the stellar fly-by(s). A single fly-by can excite the inclination of a CBP to more than 5° (PGT5 > 10 per cent). For planets close to the unstable boundary of the binary, two to three close fly-bys cause significant inclination excitation. If, like most stars, close binaries are born in clusters, our simulations suggest that there may be high-inclination planets around binaries with a_B > 1 au. Besides, it is worth mentioning that CBPs in near polar or retrograde orbits can naturally form in clusters through this mechanism.

At present, CBDs with high-inclination angles are found in binaries with a large a_B, while CBDs (and CBPs) found in close binaries are almost coplanar. Our research implies that a stellar fly-by may be one of the possible mechanisms accounting for this difference. If the nascent CBDs and their host binaries are coplanar, this difference can be explained by the fact that the CBDs in the binaries with a larger a_B are more susceptible to stellar fly-bys. Xiang-Gruess (2016) has explored the generation of highly inclined protoplanetary discs in single-star systems through a single stellar fly-by. We will discuss how this works for CBDs in the near future.

An in-depth exploration of Kepler’s existing data, as well as the ongoing Planetary Transits and Oscillations of stars (PLATO) and Transiting Exoplanet Survey Satellite (TESS) missions, is expected to find more CBPs. The distribution of the orbital inclination angles of CBPs will be the key information to understand their formation and evolution.

ACKNOWLEDGEMENTS

We thank the reviewer for valuable and insightful comments, which have improved our manuscript substantially. This work is financially supported by National Natural Science Foundation of China (Grant Nos 11573018, 11773081), CAS Interdisciplinary Innovation Team, and the Foundation of Minor Planets of Purple Mountain Observatory. Ma C.-T. also acknowledges support from the Scientific Research Projects for Universities in Shandong Province, China (J18KB101). We thank Jun-Yi Che, Shuang Liu, Yuan Yang, Zheng-Yi Xu and Bing-Xin Yan for their help with the numerical calculations.

Footnotes

1

See http://www.univie.ac.at/adg/schwarz/multiple.html.