P-TYPE PLANET–PLANET SCATTERING: KEPLER CLOSE BINARY CONFIGURATIONS

Yan-Xiang Gong

doi:10.3847/1538-4357/834/1/55

1. INTRODUCTION

To date, 11 circumbinary planets (P-type planets) have been discovered by the Kepler telescope. They are Kepler-16b (Doyle et al. 2011), -34b and -35b (Welsh et al. 2012), -38b (Orosz et al. 2012a), -47b, -47c (Orosz et al. 2012b), -47d (Hinse et al. 2015; Welsh et al. 2015), -64b (Schwamb et al. 2014; Kostov et al. 2013), -413b (Kostov et al. 2014), -453b (Welsh et al. 2015), and -1647b (Kostov et al. 2016). It is worth mentioning that the Kepler-47 system is a multi-planetary system. Such planets formed in special star configurations have aroused great interest from scientists. Studies on how they are formed (Paardekooper et al. 2012; Lines et al. 2014; Meschiari 2014; Bromley & Kenyon 2015), their habitability (Cuntz 2015), their orbital characteristics (Doolin & Blundell 2011; Martin et al. 2016), and other aspects have been widely conducted.

Pierens & Nelson (2007) found that a single circumbinary planet would migrate inward in the gas disk and would eventually be stalled in the inner disk cavitary due to disk–planet interactions. If multiple planets formed in a circumbinary disk (i.e., Kepler-47 system), convergent migration of planets would lead to planet–planet scattering (PPS, Pierens & Nelson 2008; Kley & Haghighipour 2015). PPS is thought to play an important role in sculpting exoplanet systems. It can explain the formation of hot-Jupiters (Rasio & Ford 1996; Beaugé & Nesvorný 2012), the eccentricity distribution of giant exoplanets (Chatterjee et al. 2008), and the stellar obliquity distribution of stars with hot-Jupiters (Winn et al. 2015). Even in our solar system, some clues imply that PPS may have happened (Batygin et al. 2012). Previous studies mainly focused on PPS in single star systems, while we discuss PPS of circumbinary planets in this study.

In this paper, we focus on P-type PPS phenomenon occurring in Kepler close binary configurations. The aim is to answer: (i) how PPS affects the formation and orbital configuration of these circumbinary planets, and (ii) the differences between PPS in close binary systems and in single star systems.

2. METHODS AND INITIAL CONDITIONS

We use nine Kepler close binary configurations (binaries with planets) to limit our parameter space (see Table 1). In our model, the total mass of the binary is $1{M}_{\odot }$ , with the mass ratio of the binary ( ${q}_{B}={m}_{b}/{m}_{a}$ , m_a is the mass of the bigger star) varying from 0.2 to 1.0. The initial semimajor axis (SMA) of a binary is 0.2 au, which is a typical value in Table 1. The eccentricity of a binary is taken from 0.1 to 0.5.

Table 1. Kepler Close Binary Configurations with Planets

Binary Name	Semimajor Axis	Eccentricity	Mass Ratio: q_B
Kepler-16	0.224	0.159	0.294
Kepler-34	0.229	0.521	0.974
Kepler-35	0.176	0.142	0.912
Kepler-38	0.147	0.103	0.262
Kepler-47	0.084	0.023	0.347
Kepler-64	0.180	0.218	0.247
Kepler-413	0.101	0.037	0.661
Kepler-453	0.185	0.051	0.207
Kepler-1647	0.128	0.160	0.793

Note. Data are taken from http://exoplanet.eu/.

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Circumbinary planets found by Kepler are Saturn-like planets, so we take the mass of a planet as 1M_S. In order to reduce the number of variable parameters, we consider two equal-mass planets, as also done by Ford et al. (2001) and Marzari et al. (2005). The initial eccentricities and inclinations (relative to the orbital plan of binary) of planets are $\lt {10}^{-3}$ (taking random value). According to the numerical investigations of Holman & Wiegert (1999), there is a critical SMA a_c, beyond which planetary orbits can maintain long-term stability

$\begin{eqnarray}{a}_{c}/{a}_{B} & = & 1.60+5.1{e}_{B}-2.22{e}_{B}^{2}+4.12\mu \\ & & -4.27{e}_{B}\mu -5.09{\mu }^{2}+4.61{e}_{B}^{2}{\mu }^{2},\end{eqnarray} \tag{ 1 }$

where $\mu ={m}_{b}/({m}_{a}+{m}_{b})$ , e_B is the eccentricity of the binary.

Pilat-Lohinger et al. (2003) and Funk et al. (2009) have shown that a small inclination of a planet (relative to the orbital plan of a binary) has little effect on the value of a_c. On the other hand, the tidal truncated boundary of the circumbinary disk is also near this critical SMA (Artymowicz & Lubow 1994). Indeed, circumbinary planets found by Kepler seem to cluster just outside the zone of instability (Winn et al. 2015). So, we use a_c to set the initial SMA of the inner planet. The initial semimajor axes of two planets are put as,

$\begin{eqnarray}&&{a}_{\mathrm{1,0}}=1.1{a}_{c},\,\,{a}_{\mathrm{2,0}}={a}_{\mathrm{1,0}}+{{KR}}_{H,1},\end{eqnarray} \tag{ 2 }$

where ${R}_{H,1}={[{m}_{p}/3({m}_{a}+{m}_{b})]}^{1/3}{a}_{\mathrm{1,0}}$ , ${m}_{p}={M}_{S}$ , K determines the orbital crossing time of planets. Based on our numerical test, we take K = 5.0 in this work. We are interested in PPS taking place near the unstable boundary of a close binary. We are interested because circumbinary planets found by Kepler are located just outside this boundary (for example, Kepler-16b is located at ∼ $1.1{a}_{c}$ ). On the other hand, according to the results of fluid simulation, P-type planets will migrate inward and eventually will be stalled near the boundary of instability. If another (outer) planet moves in, scattering will occur in this area. However, ${a}_{\mathrm{1,0}}$ is an important parameter, as it reflects the influence of binaries on scattering process. So the effect of ${a}_{\mathrm{1,0}}$ on our results is also discussed (see Section 3.3). The integration time of the system is 10⁶ yrs. We find that this is long enough to show the final configuration of the system. All initial phase angles were assigned randomly from 0 to 2π. As we focused on the late-stage evolution of the systems, the effect of residual gas was ignored.

We fully integrated this four-body system using the Bulirsch–Stoer integrator in our revised Mercury package (Chambers 1999). For a specific binary configuration $[{q}_{B},{e}_{B}]$ , we did 1000 runs of integrations. All integrations conserved total energy and angle momentum within 10⁻⁶ during the 10⁶ yrs of evolution. When the distance between the planet and the barycenter of a binary is larger than 100 au, the planet was removed from the system. A collision-induced merger happens when the distance between two bodies is less than the sum of their radii.

In order to compare our results with PPS in a single star system, we did a set of additional simulations (1000 runs). We adopted a similar method as that used in Ford et al. (2001). The integrations were performed for a system containing two equal-mass planets of 1M_S. The mass of the central star is $1{M}_{\odot }$ . The initial semimajor axes of the two planets are ${a}_{\mathrm{1,0}}=3$ au and ${a}_{\mathrm{2,0}}={a}_{\mathrm{1,0}}+{{KR}}_{H,1}$ , where ${R}_{H,1}={[{m}_{S}/3{M}_{\odot }]}^{1/3}{a}_{\mathrm{1,0}}$ and K = 4.5 (similar the K value used in Chatterjee et al. 2008). Each system was integrated for 10⁷ yrs, which is $2\times {10}^{6}$ times the initial period of the initial inner planet and is typically much longer than the timescale for the onset of instability (Chatterjee et al. 2008). The initial eccentricities and inclinations of planets are $\lt {10}^{-3}$ . The above choices make the simulation similar to the case of ${a}_{B}\to 0$ in our P-type PPS simulations.

3. SIMULATION RESULTS AND ANALYSIS

3.1. Overview of Results

According to the number of surviving planets, the final configurations of the system can be divided into three categories.

2P—two planets survived PPS.
1P—E: one planet was ejected; M: one planet merged with the other; C: one planet collided with the binary.
0P—E+E: two planets were both ejected; M+E: one planet merged into the other and then was ejected; C+E: one planet was ejected and the other collided with the binary.

Case C and C+E appear rarely (with a probability of $\leqslant 1 \%$ ). In Figure 1 we show the statistical distribution of the number of the surviving planet(s). In 1P systems, the branching ratios of the two cases (Ejection and Merger) are also given. In all the binary configurations we considered, 1P cases are the dominant outcomes ( $\geqslant 90 \%$ for all $[{q}_{B},{e}_{B}]$ ), with the probability of 2P cases at nearly zero. But for a typical PPS that takes place in a single star system, 2P cases are the dominant outcomes (nearly 80%) at the end of the integration. It should be noted that the final number of 2P cases in single star systems would be smaller if we integrated them for a longer time. There is no analytic criterion for Lagrange stability of two-planet systems (the orbits that are protected against either ejections or collisions with the star). Based on a large number of long-term numerical integrations, Petrovich (2015) gave an empirical boundary that best separates stable systems from systems experiencing either ejections or collisions with the star:

$\begin{eqnarray}&&\displaystyle \frac{{a}_{\mathrm{out}}(1-{e}_{\mathrm{out}})}{{a}_{\mathrm{in}}(1+{e}_{\mathrm{in}})}\gt 2.4{[\max ({\mu }_{\mathrm{in}},{\mu }_{\mathrm{out}})]}^{1/3}{\left(\displaystyle \frac{{a}_{\mathrm{out}}}{{a}_{\mathrm{in}}}\right)}^{1/2}+1.15,\end{eqnarray} \tag{ 3 }$

where ${\mu }_{\mathrm{in}}$ ( ${\mu }_{\mathrm{out}}$ ) is the planet-to-star mass ratio of the inner (outer) planets. Systems that do not satisfy this condition by a margin of $\geqslant 0.5$ are expected to be unstable. If we use this boundary to check the final orbits of the surviving planets in 2P cases, we find ∼52% of them satisfy this criterion. It means that considerable 2P cases are stable long-term.

**Figure 1.** Branching ratios of various outcomes of P-type PPS. Solid red lines: two planets survived (2P); solid blue line: one surviving planet (1P); and the solid black line: no surviving planet (0P). For case 1P, long-dashed blue line: one planet was ejected out of the system; dotted–dashed blue line: one planet merged with another. Top panel: branching ratios against the binary mass ratio q_B, ${e}_{B}=0.1$ . Bottom panel: branching ratios against the binary eccentricity e_B, ${q}_{B}=0.5$ . In contrast, typical outcomes of PPS in a single star case are given in the corresponding right panel.
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**Figure 1.** Branching ratios of various outcomes of P-type PPS. Solid red lines: two planets survived (2P); solid blue line: one surviving planet (1P); and the solid black line: no surviving planet (0P). For case 1P, long-dashed blue line: one planet was ejected out of the system; dotted–dashed blue line: one planet merged with another. Top panel: branching ratios against the binary mass ratio q_B, ${e}_{B}=0.1$ . Bottom panel: branching ratios against the binary eccentricity e_B, ${q}_{B}=0.5$ . In contrast, typical outcomes of PPS in a single star case are given in the corresponding right panel.
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If we take the same K value in the single star case K = 5, more 2P cases will appear in a single star PPS case at the end of the simulation. There were almost no 2P cases left in the binary system, indicating that a close binary plays an important role in the process of scattering. Indeed, we found that the SMA and eccentricity of the binary have a slight change in the PPS process. The binary is involved in the scattering process, which greatly increases the ratios of ejection and mergeing. As a result, more single-planet systems are produced in the close binary case.

Another new phenomenon is that considerable 0P cases appeared in P-type PPS (nearly 10%). In single star systems, at least one planet will be left in the system due to the conservation of energy. But in close binary systems, planetary systems can be completely destroyed! In our simulations, the ratios of the three 0P cases (E+E, M+E, and E+C) are about 81%, 18%, and 1%, respectively. We will discuss the details of this collective escape in the future.

Figure 1 also shows that the mass ratio of the binary, q_B, has little effect on the results. It looks as if the probability of 0P decreases with the increase of e_B in Figure 1. We speculate that it is related to the choice of the initial locations of the planets. The initial position ${a}_{\mathrm{1,0}}$ is a function of e_B. With the increase of e_B, the initial position gets farther away from the binary, so the influence of the binary on the scattering process is weakened. As a result, the number of such extreme cases (0P) is reduced.

3.2. Final Orbital Elements of Planets

For 1P cases, we investigated the final SMA and eccentricity of the surviving planet. Figures 2 and 3 show the distributions of the semimajor axes and eccentricities of planets with different binary mass ratios $[{q}_{B}\,=\,$ (0.2, 0.6, and 1.0), e_B = 0.1] and eccentricities [q_B = 0.5, e_B = (0.1, 0.3, and 0.5)], respectively. We found the peak value of the eccentricities of the surviving planets to increase with the increase of e_B, and decrease with the increase of q_B (ejection case). Besides this feature, there is no obvious difference in the eccentricity distribution for different q_B or e_B. However, there are obvious differences between PPS in the single star case and the close binary case.

**Figure 2.** Final semimajor axis and eccentricity distributions of surviving planets in 1P cases. The left two panels are for the ejection case, and the right two panels are for the merging case. The short-dashed black lines, the long-dashed red lines, and the solid blue lines are for the q_B = 0.2, 0.6, and 1.0 cases ( ${e}_{B}=0.1$ ), respectively. In contrast, eccentricity distribution of a single star case is also plotted in the corresponding panel (thin black lines).
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**Figure 3.** Final semimajor axis and eccentricity distribution of surviving planets against the eccentricity of a binary. The short-dashed black lines, the long-dashed red lines, and the solid blue lines are for the e_B = 0.1, 0.3, and 0.5 cases ( ${q}_{B}=0.5$ ), respectively. Other conventions are the same as in Figure 2.
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**Figure 3.** Final semimajor axis and eccentricity distribution of surviving planets against the eccentricity of a binary. The short-dashed black lines, the long-dashed red lines, and the solid blue lines are for the e_B = 0.1, 0.3, and 0.5 cases ( ${q}_{B}=0.5$ ), respectively. Other conventions are the same as in Figure 2.
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(i) Low eccentricities. In the single star case, the eccentricity of the surviving planet is in the range of 0.4–0.8 (Ford et al. 2001), with a median value around 0.6 (see also Chatterjee et al. 2008). In the close binary case, the final eccentricities are generally low. This is because the angular momentum (AM) exchange between planets is insufficient if PPS takes place near the instable boundary of the binary. The simulation shows that after a few close encounters between the planets, the inner planet loses a small amount of AM as it approaches the binary, and then soon it is scattered out of the system by the binary (a typical example is plotted in Figure 4). By examining the evolution map, we found that when one planet is ejected, the semimajor axes and eccentricity of the binary dramatically change. This means that, in the entire process, AM exchange takes place mainly between the binary and the ejected planet. As a result, the surviving planet gets little AM or a low eccentricity. In the single star case, sufficient close encounters between planets enable the surviving one to get enough AM, and it therefore has a high eccentricity. Another interesting phenomenon is that the ejected planet is mainly the inner one (about 70% in ejection cases) in the close binary case, as we can see in Figure 4. However, in the single star case, after close encounters, two equal-mass planets are dynamically indistinguishable. We cannot predict which planet will be scattered out of the system.

**Figure 4.** Time evolution of semimajor axis (a), pericenter (q), and apocenter (Q) distances of two planets. Upper panel: a typical ejection process in the P-type PPS case where the initial inner planet is ejected out of the system at ∼29,437 yr. The changes of a_B and e_B are also plotted. The dashed red line denotes the corresponding a_c derived by Holman & Wiegert (1999). Lower panel: a typical ejection process in the single star case where the initial outer planet is ejected out of the system at ∼3,416,369 yr.
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(ii) Inward migration versus outward migration. In the single star case, the surviving planet will migrate inward due to the loss of energy. The ejected planet typically leaves the system with a small positive energy (Moorhead & Adams 2005), therefore the final SMA of the surviving planet can be estimated by energy conservation (see Ford et al. 2001; Ford & Rasio 2008). It is possible for the inner planet (if it is the surviving one) to shrink almost half of its initial orbit for equal-mass planet scattering. We might as well call it inward migration scattering. But in the close binary case, the surviving planet is the one that has migrated outward as we see in Figure 4. Moreover, we can see ${a}_{f}/{a}_{\mathrm{1,0}}\gt 1$ in Figure 2 for the ejection case, which means that even if the ejected planet was initially the outer one, then the inner planet has since migrated outward after PPS. This scattering pattern is unique for P-type planets.

3.3. The Effect of ${a}_{\mathrm{1,0}}$ on the Result

The instability growth timescale of a circumbinary planet system depends not only on the initial relative distance between planets, but also on the initial relative distance between the planets and inner binary. Far away from the binary, the influence of the binary is weakened. PPS will be different from the above-mentioned results. We discuss the effect of ${a}_{\mathrm{1,0}}$ on scattering results in this section. We take ${e}_{B}=0.1$ , the mass ratio ${q}_{B}=0.2$ , and K = 4.5. Other parameters are as before, while the value of f_c is changed ( ${f}_{c}=1.1,2.1,3.1$ ). The integration time of ${f}_{c}=2.1,3.1$ cases is 10⁷ yrs. For each f_c, we integrated 500 systems. The results are shown in Figure 5.

On the top panel of Figure 5, branching ratios are given. We can see that, with the increase of ${a}_{\mathrm{1,0}}$ , more 2P cases appear and the probability of 1P gets smaller. Another thing to note is that the probability of 0P is nearly zero. The results tend toward the scattering outcomes of single star systems. In the bottom panel of Figure 5, we find that eccentricity distribution is related to ${a}_{\mathrm{1,0}}$ . Near the binary, the final eccentricities are generally low. Far away from the binary, the eccentricities are generally high. The dispersion of the eccentricity distribution is also related to ${a}_{\mathrm{1,0}}$ , which reflects the competition between planet–planet and binary–planet interaction.

It is noteworthy that the two planets were identical in our model. Ford et al. (2001) found that dynamical instabilities of two equal-mass planets overproduced highly eccentric orbits, which caused a discrepancy between simulations and the observed orbits of exoplanets. To solve this problem, Ford & Rasio (2008) proposed PPS between unequal-mass planets. Dynamic interactions tend to eject preferentially the less massive planet in an unequal-mass model. As a result, the more massive one remains and acquires a significant eccentricity, which depends on the mass ratio of the two planets. Therefore, the unequal-mass model can easily reproduce the observed eccentricity distribution by assuming a plausible mass distribution (see also Chatterjee et al. 2008; Jurić & Tremaine 2008; Raymond et al. 2008).

From the statistical point of view, even if the two planets have comparable masses in the circumbinary case, if PPS takes place at different locations in different systems, then the observed eccentricity distribution of circumbinary planets will be diverse. In addition, the maximum eccentricity is no more than 0.8 in single star systems, which is independent of the distribution of the planet mass ratios (Ford & Rasio 2008). While in the circumbinary case, the maximum eccentricity can exceed 0.9. Additional contributions of the AM and energy come from the inner binary.

4. SUMMARY AND DISCUSSION

In this paper, we have discussed P-type planet scattering occurring in Kepler close binary configurations. Due to the perturbations of the inner binary, many new features appear in the P-type planet scattering process. If PPS takes place near the location of the currently discovered circumbinary planets, then scattering presents some new features, which are summarized as follows. First, a close binary has a significant influence on the scattering process. About 10% of the system has been completely destroyed in our simulation. In more than 90% of the systems, there is only one planet left. Second, the surviving planets generally have a lower eccentricity. Third, the ejected planet is generally the innermost of the two initial planets. If the initial positions of the planets are moved away from the binary, the result statistically tends toward those around single stars. In this process, the competition between two kinds of interactions (binary–planet and planet–planet) makes the eccentricity distribution of surviving planets diverse.

Binary perturbations make the formation of circumbinary planets difficult. For example, the tidal truncation of the circumbinary disk, the difficulty of the formation of embryos in the mass-erosion region, and the short life of the circumbinary gas disk are all detrimental factors to the formation of circumbinary planets. Our simulation suggests that PPS is another negative factor in the formation of these planets. Another finding of our work is that we cannot deduce that these Kepler circumbinary planets are the survivals of mild gas-disk migration by only their low eccentricities. They may experience scattering processes. In fact, not all of the Kepler circumbinary planets are in the vicinity of the unstable boundary. For example, Kepler-47c is located at 5.0a_c, and Kepler-1647b is located at 7.4a_c. Are they the surviving planets that have migrated outward after a PPS process? Such a question requires more observations to answer. In addition, it also needs further explaination as to why Kepler-47c might have an eccentricity as high as 0.4 (Orosz et al. 2012b). PPS may be worth some thought.

We thank the anonymous referee for constructive comments and suggestions. This work is supported by the Natural Science Foundations of China (Grants 11573018 and 11333002) and the Shandong Provincial Natural Science Foundation, China (ZR2014JL004). We also acknowledge the support from Taishan University (Doctoral Fund Y-01-2014021). We thank Zhou Ji-lin and Xie Ji-wei for their helpful discussions.

P-TYPE PLANET–PLANET SCATTERING: KEPLER CLOSE BINARY CONFIGURATIONS

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ABSTRACT

1. INTRODUCTION

2. METHODS AND INITIAL CONDITIONS

3. SIMULATION RESULTS AND ANALYSIS

3.1. Overview of Results

3.2. Final Orbital Elements of Planets

3.3. The Effect of ${a}_{\mathrm{1,0}}$ on the Result

4. SUMMARY AND DISCUSSION

P-TYPE PLANET–PLANET SCATTERING: KEPLER CLOSE BINARY CONFIGURATIONS

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Permissions

Share this article

Author e-mails

Author affiliations

ORCID iDs

Dates

ABSTRACT

1. INTRODUCTION

2. METHODS AND INITIAL CONDITIONS

3. SIMULATION RESULTS AND ANALYSIS

3.1. Overview of Results

3.2. Final Orbital Elements of Planets

3.3. The Effect of {a}_{\mathrm{1,0}} on the Result

4. SUMMARY AND DISCUSSION

3.3. The Effect of ${a}_{\mathrm{1,0}}$ on the Result