The Scattering Outcomes of Kepler Circumbinary Planets: Planet Mass Ratio

Thus far over 3400 planets have been discovered by the Kepler space telescope (Borucki et al. 2010; Lissauer et al. 2011; Batalha et al. 2013; Mazeh et al. 2013; Fabrycky et al. 2014; Wang et al. 2015). These planets show a variety of orbital configurations, which has greatly improved our understanding of planetary formation (Dong & Ji 2013; Lee et al. 2013; Jin et al. 2014; Raymond & Cossou 2014; Wang & Ji 2014; Batygin et al. 2016; Dong et al. 2017). For example, many systems contain multiple small planets with orbital periods less than ~50 days, known as tightly spaced inner planets (MacDonald et al. 2016), while plenty of planets are found in or near mean-motion resonances within these systems (Wang et al. 2012; Lee et al. 2013; Wang & Ji 2014, 2017; Zhang et al. 2014; Marti et al. 2016; Mills et al. 2016; Sun et al. 2017).

One of the most exciting findings of Kepler is the discovery of several circumbinary planets around main-sequence stars. Due to the perturbation of the inner binary, their formation, orbital characteristics, and habitability of these special bodies bring new challenges to planetary science. At present, 11 circumbinary planets have been discovered by Kepler belonging to 9 planetary systems (http://exoplanet.eu/). Among them, Kepler-47 is a multiple-planet system (Orosz et al. 2012; Hinse et al. 2015; Welsh et al. 2015).

The masses and orbital configurations of these planets are listed in Table 1. The a_c in Table 1 is the critical Semimajor Axis (SMA) of a planet (relative to the barycenter of the binary) beyond which planetary orbits can maintain long-term stability, except for the unstable islands associated with N:1 resonance with the binary (Holman & Wiegert 1999).

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

where $\mu ={m}_{B}/({m}_{A}+{m}_{B})$ is the mass ratio of the binary, and e_B and a_B are its eccentricity and SMA, respectively. Several characteristics of these planets are noteworthy. (1) The binary and planetary orbits are aligned within a few degrees. The highest relative inclination is 25 (Winn et al. 2015). (2) Except Kepler-1647b, most of them cluster just outside of the zone of instability. (3) The majority of them have a nearly circular orbit. Despite some of the above trends, the specter of selection effects lurk, which are consistent with the results predicted by disk-driven migration models (Pierens & Nelson 2007, 2013; Kley & Haghighipour 2014, 2015).

Table 1.  Mass and Orbital Configuration of Kepler Circumbinary Planets

Planet Name	Mass (MJ)	Semimajor Axis (au)	Eccentricity	Forced Eccentricitya	${a}_{p}/{a}_{c}$
Kepler-16b	0.33	0.705	0.007	0.034	1.16
Kepler-34b	0.22	1.090	0.182	0.002	1.31
Kepler-35b	0.13	0.603	0.042	0.002	1.21
Kepler-38b	0.38	0.464	<0.032	0.024	1.25
Kepler-47b	⋯	0.296	<0.035	0.004	1.48
Kepler-47c	⋯	0.989	<0.411	⋯	4.95
Kepler-64b	0.53	0.643	0.054	0.044	1.26
Kepler-413b	0.21	0.355	0.118	0.003	1.40
Kepler-453b	0.03	0.788	0.038	⋯	1.85
Kepler-1647b	1.52	2.721	0.058	⋯	7.41

Note. Data in columns 2–4 are taken from http://exoplanet.eu/.

^aBromley & Kenyon (2015).

Download table as:  ASCIITypeset image

Did these circumbinary planets undergo planet–planet scattering (PPS) processes? Recent studies provide some clues to this issue. Bromley & Kenyon (2015) examined the orbital characteristics of circumbinary planets in the context of current planet formation scenarios. They found that the forced eccentricity at Kepler-34b's location is low (about 0.002), but the eccentricity of the observed planet is much larger (~0.18). Such high free eccentricity and a high mass of the planet favor a scattering process. For comparison, Kepler-413b has a significant free eccentricity about 0.12. However, its forced eccentricity is only 0.003. The large free eccentricity of Kepler-413b tends to preclude the migrate-in-gas mode (Bromley & Kenyon 2015). Its orbital configuration is consistent with scattering events.

Kley & Haghighipour (2015) considered the disk-driven migration of Kepler-34b using two-dimensional hydrodynamical simulations. They found that the planet's final equilibrium position lies beyond the observed location of Kepler-34b. To account for the closer orbit of Kepler-34b, they proposed a scenario in which there are two planets in the system. The convergent migration of the two planets often leads to capture into mean-motion resonances (MMR). A weak planet–planet scattering process ensues when the inner planet orbits inside the gap of the disk. The above-mentioned scenario can reproduce the observed orbit of Kepler-34b. In this model, another planet would still reside in the system on a long-period orbit (~1.5 au), which may fail to transit the central binary during the operation of Kepler as they suggested.

In this work, we extensively explored the scattering outcomes of circumbinary planets, focusing on the effects of planet mass ratio. In a single-star system, the planet mass ratio is a key parameter in the dynamical scattering process (Ford & Rasio 2008). A two equal-mass planets scattering model gives a narrow distribution of final eccentricities, which cannot reproduce the eccentricity distribution of the observed giant planets. However, a two unequal-mass planet scattering model predicts a broader range of final eccentricities. With a reasonable distribution of planet mass ratios, the observed eccentricities can be reproduced (Chatterjee et al. 2008; Ford & Rasio 2008). However, how does planet mass ratio affect the scattering outcomes of circumbinary planets? In Gong (2017), we investigated the scattering process of two equal-mass planets by considering the role of binary configurations (i.e., μ and e_B). We explored all kinds of close binary configurations and showed some new features of scattering events, which differ from the scattering results revealed in single-star systems. Herein, based on a two unequal-mass planet scattering model, we make an extensive study of scattering outcomes of circumbinary planets through numerical simulations, then explore the final statistical configurations of the surviving planets after PPS, with the aim to understand the formation of currently observed Kepler circumbinary companions.

Scattering events have been found in the hydrodynamic simulations of multiple planets in a circumbinary disk. For example, disk-driven migration of multiple low-mass circumbinary planets (5–20 Earth mass) in an artificial binary system were discussed in Pierens & Nelson (2008). They showed that two planets usually undergo dynamical scattering for mass ratio $q={m}_{\mathrm{inner}}/{m}_{\mathrm{outer}}\lt 1$. For $q\gt 1$, the planets will be finally locked into MMRs. As stated previously, Kley & Haghighipour (2015) further showed that two circumbinary planets can be captured into MMR as a result of inward convergent migration. In the subsequent process, the planets usually undergo dynamical scattering (Kley & Haghighipour 2015).

In this work, we model the late evolution stage of circumbinary systems by ignoring the effect of the residual gas (Chatterjee et al. 2008; Jurić & Tremaine 2008; Raymond et al. 2008; Beaugé & Nesvorný 2012; Moeckel & Armitage 2012). Moeckel & Armitage (2012) testified that this is a reasonable approximation. The hydrodynamic outcomes of planet scattering in transitional disks are discussed in their work. They showed that N-body dynamics and hydrodynamics of scattering into one-planet final states are nearly identical. The eccentricity distributions of the surviving planets are almost unaltered by the existence of the residual gas.

The article is organized as follows. Section 2 presents our numerical model and initial conditions. In Section 3, we give the numerical results and the analyses. Section 4 discusses the probability of reproducing the orbital configurations of Kepler-34b and Kepler-314b. Finally, Section 5 summarizes the major results and discusses the orbital evolution theory of circumbinary planets.

We start our scattering simulations with two planets that have been extensively studied in single-star systems (see Ford & Rasio 2008 and the references therein). Another advantage of the two-planet model is that understanding this simple case facilitates the analysis of simulations with more planets. Gong (2017) showed that binary configurations have no substantial effect on the scattering results. Therefore, we take a Kepler-16(AB)-like close binary configuration (Doyle et al. 2011) as a baseline. That is, the SMA, eccentricity, and mass ratio of the binary are ${a}_{B}=0.22\,\mathrm{au}$, ${e}_{B}=0.16$, ${q}_{B}={M}_{b}/{M}_{a}=0.29$, respectively. The total mass of the binary is 1 M_⊙.³ According to the mass distribution of the circumbinary planets (see Table 1), we consider five sets of mass combinations of the planets. We found that, in addition to the mass ratio, the initial relative position of the two planets also affects the simulation results. A bracket is used to indicate the initial relative position of the two planets. For example, [M_S, M_J] refers to the initial inner/outer planet: a Saturn-like/Jupiter-like planet. All of the mass combinations are given in Table 2.

Table 2.  Ejection Preference of the Two Circumbinary Planets

Mass Ratio	Planets	1.2ac		2.0ac		Single-star Systema
Ejection Preference		Ejein	Ejeout	Ejein	Ejeout	Ejein	Ejeout
q = 1	$[{M}_{J},{M}_{J}]$	0.39	0.15	0.25	0.19	⋯	⋯
q = 0.5	$[0.5{M}_{J},{M}_{J}]$	0.46	0.08	0.42	0.13	⋯	⋯
	$[{M}_{J},0.5{M}_{J}]$	0.36	0.21	0.19	0.33	⋯	⋯
q = 0.3	$[{M}_{S},{M}_{J}]$	0.44	0.03	0.62	0.01	0.13	0.00
	$[{M}_{J},{M}_{S}]$	0.24	0.22	0.03	0.54	0.00	0.17
q = 0.15	$[0.5{M}_{S},{M}_{J}]$	0.45	0.01	0.54	0.00	⋯	⋯
	$[{M}_{J},0.5{M}_{S}]$	0.12	0.35	0.00	0.53	⋯	⋯
q = 0.03	$[0.1{M}_{S},{M}_{J}]$	0.65	0.00	0.53	0.00	⋯	⋯
	$[{M}_{J},0.1{M}_{S}]$	0.00	0.69	0.00	0.52	⋯	⋯

Note. Eje_in (Eje_out) is the fraction of the initial inner (outer) planets that were ejected out of the systems, in total 1000 runs.

^aA set of PPS simulations (1000 runs) in single-star systems is performed for comparison. The initial SMAs of the two planets are ${a}_{\mathrm{1,0}}=3\,\mathrm{au}$, ${a}_{\mathrm{2,0}}={a}_{\mathrm{1,0}}+{{KR}}_{\mathrm{Hill},m}$, K = 3. The initial distance between the two planets is close to their Hill stability boundary.

Download table as:  ASCIITypeset image

Among these Kepler circumbinary planets, Kepler-1647b has the longest orbital period and is located at $7.4\,{a}_{c}$ (2.7 au; Kostov et al. 2016). The planet may not undergo significant disk-driven migration. Kepler-47c is the outer planet of a multiple-planet system. Except for these, the majority of Kepler circumbinary planets lie between $1.16\,{a}_{c}$ and $1.85\,{a}_{c}$. Hence, we use this interval to set the initial position of the inner planet. For the initial SMA of the inner planet, we consider two cases ${a}_{\mathrm{1,0}}=1.2\,{a}_{c}$ and $2.0\,{a}_{c}$, respectively. They represent the scattering occurring near the stable boundary of the binary or away from it. The initial SMA of the outer planet is

$$\begin{eqnarray}&&{a}_{\mathrm{2,0}}={a}_{\mathrm{1,0}}+{{KR}}_{\mathrm{Hill},m},\end{eqnarray} \tag{ 2 }$$

where ${R}_{\mathrm{Hill},m}$ is the mutual Hill radius defined as

$$\begin{eqnarray}&&{R}_{\mathrm{Hill},m}={\left(\displaystyle \frac{{m}_{1}+{m}_{2}}{3{M}_{* }}\right)}^{1/3}\left(\displaystyle \frac{{a}_{1}+{a}_{2}}{2}\right).\end{eqnarray} \tag{ 3 }$$

K is an important parameter that may affect the unstable timescale of the system (Chatterjee et al. 2008; Kratter & Shannon 2014). A compromising strategy is taken in choosing the K value in this work. We avoid unphysical very closely packed systems (small K). On the other hand, we do not take a large K because the required integration time is too long to perform a large-sample statistical study.

For an isothermal and radiative disk, Kley & Haghighipour (2015) revealed that planets typically result in the capture of low-order MMR with period ratios of 3:2, 5:3, 2:1, etc. At ${a}_{\mathrm{1,0}}=1.2\,{a}_{c}$, we take K = 4. The resultant initial distance of planet is larger than 3:2 and near 5:3 resonance (K = 3.4 and 4.3, respectively) in our model. In a single-star system, the unstable timescale of two-planet systems can be measured using Hill stability criteria (Gladman 1993). For a $[{M}_{S},{M}_{J}]$ system with ${a}_{\mathrm{1,0}}=3$ au in a singe star system, the Hill stability criteria gives $K\sim 3$ (the unit is the mutual Hill radius). Thus, in the binary system, we take K = 3 for ${a}_{\mathrm{1,0}}=2.0\,{a}_{c}$ (scattering taking place away from the binary). Combined with our numerical tests, we integrate each system up to 10⁶ years for ${a}_{\mathrm{1,0}}=1.2$, and 10⁷ years for ${a}_{\mathrm{1,0}}=2.0\,{a}_{c}$. Numerical tests showed these integration times are long enough to reflect the scattering process. The initial eccentricities and inclinations of planets are $\lt {10}^{-3}$. All initial phase angles were assigned randomly from 0 to 2π. We fully integrated each system using the Bulirsch–Stoer integrator in our revised Mercury package (Chambers 1999). For every mass combination and the different ${a}_{\mathrm{1,0}}$ (see Table 2), we perform 1000 runs. The type is referred to as "ejections" meaning the distance between the planet and the barycenter of a binary is larger than 500 au.

In this work, we assume that the currently observed single-planet systems are the products of PPS of original multiple-planet systems. Thus, we focus on the resulting single-planet systems. Some of them are the merger of two planets. Considering the mass and angular momentum conservation, the new planet generally has a larger mass and a low eccentricity, which is similar to the results of PPS in single-star systems (Ford et al. 2001; Ford & Rasio 2008). The majority of single-planet systems come from a scenario in which one planet is ejected out of the system. In the following, we analyze these systems in detail.

3.1. Mass Ratio Versus Ejection Preference

In single-star systems, the ejections originate from the less massive one of the two planets, regardless of whether it was initially the inner or the outer planet (Ford & Rasio 2008). For circumbinary planets, this conclusion is conditional. It depends on the planet mass ratio q ($\lt 1$) and the initial relative location of the two planets (see Table 2). Gong (2017) found that for equal-mass planets, the initial inner planets are peculiarly prone to be ejected if PPS takes place near the unstable boundary of the binary. This trend is maintained as long as the mass ratio of the planets is greater than 0.3, as we can see in Table 2 where ${a}_{\mathrm{1,0}}=1.2\,{a}_{c}$.

However, as the mass ratio becomes smaller, this tendency disappears. For q = 0.03, the less massive planets are always scattered out of the system, regardless of their initial relative positions. Our numerical simulation suggests that if the planetary mass ratio is greater than ~0.3, the initial inner planets are more likely to be ejected. However, if the mass ratio is less than 0.3, the less massive planets are highly likely to be scattered out of the system. This position dependency does not exist if the initial locations of the two planets are moved away from the binary (${a}_{\mathrm{1,0}}=2.0\,{a}_{c}$). Regardless of their initial positions, the ejections are of the less massive planets.

Kepler-34b and Kepler-413b have a mass of $\sim 0.2{M}_{J}$. We carried out additional simulations to explore how the total mass of two planets affects the above results. We set the mass of the more massive planet to be $0.2{M}_{J}$, and five mass ratios q = 1, 0.5, 0.3, 0.15, 0.03 were considered. In addition to K, the initial distance between two planets is also relevant to their total mass (see Equation (2)). Through numerical examination, we adopted K = 5 to avoid very closely packed systems. The other parameters remain unchanged. The results for ${a}_{\mathrm{1,0}}=1.2{a}_{c}$ are shown in Table 3. As can be seen from Table 3, although the fraction is slightly different, the general trends of the ejection preferences are similar to each other. This indicates that the total mass has little effect on the outcomes, at least for giant planets with mass ${m}_{p}\approx 0.2{M}_{J}$.

Table 3.  Ejection Preference of Two Less Massive Planets

Mass ratio	Planets	1.2ac
Ejection Preference		Ejein	Ejeout
q = 1	$[0.2{M}_{J},0.2{M}_{J}]$	0.52	0.19
q = 0.5	$[0.1{M}_{J},0.2{M}_{J}]$	0.55	0.06
	$[0.2{M}_{J},0.1{M}_{J}]$	0.37	0.26
q = 0.3	$[0.06{M}_{J},0.2{M}_{J}]$	0.69	0.02
	$[0.2{M}_{J},0.06{M}_{J}]$	0.35	0.32
q = 0.15	$[0.03{M}_{J},0.2{M}_{J}]$	0.72	0.01
	$[0.2{M}_{J},0.03{M}_{J}]$	0.29	0.41
q = 0.03	$[0.006{M}_{J},0.2{M}_{J}]$	0.70	0.00
	$[0.2{M}_{J},0.006{M}_{J}]$	0.00	0.69

Download table as:  ASCIITypeset image

As previously mentioned, several studies have shown that planets born in a circumbinary disk will migrate inward and eventually be stalled near the inner hole of the disk (Pierens & Nelson 2008; Kley & Haghighipour 2015). If an outer planet migrates toward it, PPS will occur. If we assume that the mass power law of the circumbinary planets formed in a system is the same as that of the solar system, the inner planet (Jupiter) is more massive than the outer planet (Saturn). Thus, our results imply there is an equivalent or even a larger probability for the inner more massive planets to be ejected out of the system if their mass ratio is larger than a critical value. The currently observed Kepler circumbinary planets are generally less massive. Whether PPS occurring in the late stage is a possible mechanism accounting for this phenomenon, which should be examined by future observations and investigations.

3.2. Mass Ratio Versus Orbital Element Distribution

We discuss the final orbital distribution of the surviving planets in two subsets (${a}_{\mathrm{1,0}}=1.2\,{a}_{c}$ and ${a}_{\mathrm{1,0}}=2.0\,{a}_{c}$). Figure 1 shows the eccentricity distributions of the surviving planets for ${a}_{\mathrm{1,0}}=1.2\,{a}_{c}$. To show the details, we present the results according to different mass ratios and initial relative positions. In the top panel of Figure 1, the final eccentricity distributions are of the initial inner planets that survive the PPS. The bottom panel shows the eccentricity distribution of the initial outer planets that survive the PPS. For clarity, we only draw the cases of q = 0.3 and 0.03. From Figure 1, we find that if the mass ratio of the planets is small q = 0.03, the remaining planets maintain a small eccentricity. The values are roughly equivalent to their forced eccentricities. This is because in the scattering process, the less massive planet was scattered out of the system quickly under the combined actions of the massive one and the binary. As a result, the more massive planet gets little angular momentum, so it maintains a small eccentricity. An example is shown in Figure 2.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For q = 0.3, to gain a lower eccentricity, the surviving planets must be the massive ones of the two initial planets, regardless of which planet was initially closer (see the red and black line on the top and bottom panel, respectively). Conversely, when the surviving planets are the less massive ones, their eccentricities are generally large with a median value 0.6. An example is presented in Figure 3. Besides, the range of eccentricity is related to the initial relative position of the less massive planet. If the planet is the initial inner one, its final eccentricity is larger than the initial outer one, indicating that it gets more angular momentum in a more violent dynamical process. It seems that the eccentricity distribution of the outer surviving less massive planet is more diffuse than the inner one. If we assume that the currently observed circumbinary planet is the product of PPS, the value of its eccentricity can be used to estimate the mass ratio of the original two planets.

Zoom In Zoom Out Reset image size

Gong (2017) found that, after PPS, the SMA of the surviving planets increases, contrary to the scattering phenomena in single-star systems. In order to study the SMA variation of the remaining planets, we plotted the distribution of ${a}_{p}/{a}_{p,0}$ (the ratio of the final SMA of the surviving planet to its initial SMA) in Figure 4. If the ratio is greater than 1, the planet is shifted outward after PPS. We find that, nearly in all cases, the SMAs of the surviving planets are $\gt 1$ for ${a}_{\mathrm{1,0}}=1.2\,{a}_{c}$ cases. The results are not related to the planet mass ratio and the initial relative position of the two planets. For q = 0.3, the maximum ${a}_{p}/{a}_{p,0}$ can reach ~15.

Zoom In Zoom Out Reset image size

For ${a}_{\mathrm{1,0}}=2.0\,{a}_{c}$, the eccentricity distribution does not significantly change (see Figure 5). But in the SMA distribution, a double peak structure emerges, as we can see in Figure 6. This suggests that some of the surviving planets migrated inward during PPS, which is a typical feature of the scattering in single-star systems. It indicates that at $2.0\,{a}_{c}$, scattering begins to appear, characteristic of PPS in single-star systems. The influence of the inner binary is weakened at this location. In Figure 7, we show that the SMA of a surviving planet shrinks after PPS. Our additional numerical simulations showed that if the scattering occurs in the region of [1.2–1.8]a_c, the scope of most currently discovered Kepler circumbinary planets, almost all of the SMAs of the surviving planets are incremented after PPS. This suggests that if these planets were the survivors of PPS, their initial location would have been closer than their currently observed value.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Moreover, we found that the surviving planet generally maintains a nearly coplanar configuration with the binary. The distribution of the inclination is also related to the mass ratio and the initial relative position of the two planets. The inclination distribution of the surviving planets is related to the initial inclination of the two planets, which we will discuss in future works.

4.1. Kepler-34b

The orbital evolution of Kepler-34b in a protoplanetary disk has been studied in several works. Pierens & Nelson (2013) considered the migration and gas accretion scenarios of Kepler-34b. For the fully formed planet, its final location is determined by the adopted physical parameters of the disk (aspect ratio h, viscous stress parameter α, etc.). In addition, they took into account low-mass cores that migrate and accrete gas while the disk is being dispersed. In most cases, the planets halt migration beyond the currently observed orbit of Kepler-34b (1.09 au). However, in the case of h = 0.05 and $\alpha ={10}^{-4}$, the planets can migrate across the Kepler-34b's currently observed orbit and reach ~1 au with a nearly zero eccentricity. Recently, Mutter et al. (2017) considered the role of self-gravity in sculpturing the structure of circumbinary disks. They showed that the scale of the inner cavity depends on the disk mass. An enhanced disk mass will cause the outer edge of the cavity to be closer to the binary. It may imply that circumbinary planets that formed in the disks can migrate to much closer regions.

Using a more sophisticated disk model, Kley & Haghighipour (2015) revisited the evolution of Kepler-34b. In their work, they further indicated that the planet stalls beyond the observed regime of Kepler-34b. To account for the closer orbit of Kepler-34b, they modeled a two-planet scenario in the simulations. They showed that the two planets can enter a 3:2 MMR and then undergo sequential weak scattering events. As a result, the inner planet can move toward the present orbit of Kepler-34b. The model may imply that there is an additional planet in the system (~1.5 au or 1.8 a_c). However, in our model, we assume that the currently observed single-planet system could simply be the survival of PPS from an original two-planet system. After PPS, one planet is completely ejected out of the system. One of our key goals herein is to observe whether this model can reproduce the current orbital configuration of Kepler-34b. In the following, we will discuss this scenario.

As previously mentioned, if the planets migrate inward and stall beyond the presently observed orbit of Kepler-34b, the ejection model cannot reproduce its current orbital configuration. After PPS, the surviving planet will migrate outwards, leading to a larger SMA. However, if Kepler-34b ever migrated to a closer location as discussed in Pierens & Nelson (2013), PPS can reproduce the observed orbital configuration of Kepler-34b.

We set the initial orbital elements of the inner planets according to Pierens & Nelson (2013) (for h = 0.05 and $\alpha ={10}^{-4}$), where ${a}_{\mathrm{1,0}}\approx 1\,\mathrm{au}$, ${e}_{\mathrm{1,0}}\approx 0.01$. The SMA of the outer planet is given according to the 3:2 MMR region with respect to the inner planet. We give the eccentricity of the outer planets an estimated value, referring to the other case in Pierens & Nelson (2013), but our result does not depend on this value. Next, we performed a number of simulations, one of which is shown in Figure 8. From Figure 8, we show that a two planets ejection model can reproduce the observed orbital configuration of Kepler-34b through the disk-driven migration plus a PPS model. Using the given parameters and different phase angles, we performed 1000 runs. Among 399 systems, one planet was ejected out of the system and the other finally survived, whereas in 126 systems, two planets merged to form one large planet. In 18 systems, no planet remained. The remaining systems had two planets at the end of the integration. We found that in 20 systems, the surviving planet had a similar eccentricity and SMA (with an error bar) of Kepler-34b. Therefore, the likelihood of producing a Kepler-34b-like planet in our model is 20/399 ≈ 5.01%.

Zoom In Zoom Out Reset image size

4.2. Kepler-413b

Kepler-413(AB) is a K+M eclipsing binary with ${a}_{B}=0.101\,\mathrm{au}$ (Kostov et al. 2014). The mass of the two dwarfs are ${m}_{A}=0.82\,{M}_{\odot }$ and ${m}_{B}=0.54\,{M}_{\odot }$, respectively. A Neptune-sized circumbinary planet, Kepler-413b, was discovered in this system on an eccentric orbit with ${a}_{p}=0.355\,\mathrm{au}$, ${e}_{p}=0.12$. At present, there is no disk migration model of Kepler-413b. Hydrodynamical simulations showed that the eccentricity of the binary has a decisive influence on the size and structure of the disk's inner cavity, and the final position of a planet depends on this size. For binary systems with nearly circular orbits, planets forming farther out in the calmer environment of the disk can migrate toward the unstable boundary of the binary (Kley & Haghighipour 2014). The Kepler-413(AB) binary has a small eccentricity (${e}_{B}=0.037$). It seems unlikely that Kepler-413(AB) can open a wide inner hole in the disk as the highly eccentric binary Kepler-34(AB) with ${e}_{B}=0.52$ (Kley & Haghighipour 2015). Thus, we assume a planet born in the Kepler-413(AB) system can migrate toward the innermost region of the disk, like Kepler-16b ($\sim 1.2\,{a}_{c}$).

Herein, we take ${a}_{\mathrm{1,0}}=1.2\,{a}_{c}$. Similarly, the initial orbit of the outer planet is set according to the 3:2 MMR location. Next, we carried out a set of simulations to investigate the orbital configuration of Kepler-413b. A typical case is illustrated in Figure 9. The initial orbital parameters of the two planets are ${a}_{\mathrm{1,0}}=0.31\,\mathrm{au}$, ${e}_{\mathrm{1,0}}=0.07$, ${a}_{\mathrm{2,0}}=0.406\,\mathrm{au}$ and ${e}_{\mathrm{2,0}}=0.02$, respectively. The masses of two planets are ${m}_{\mathrm{inner}}=0.211\,{M}_{J}$ (Kepler-413b) and ${m}_{\mathrm{outer}}=0.09\,{M}_{J}$. At $\sim 2\times {10}^{6}$ years, the outer less massive planet was ejected out of the system. From the simulations, we found that a two-planet ejection model may work to generate the final orbital configuration of Kepler-413b as currently observed. This suggests that both the eccentricity and SMA of Kepler-413b can be well reproduced with a two-planet ejection model. Using the given parameters, we carried out 1000 runs for this system by varying the phase angles. Among 422 runs, one planet was ejected and the other planet survived in the system, while for 211 systems, two planets merged into one planet. In 16 systems, there was no planet left. The remaining systems were observed to occupy two planets when the simulations were complete. We found that in 41 systems, the surviving planet bears a similar eccentricity and SMA to those of Kepler-413b. Hence, we have conclude that the likelihood of yielding a Kepler-34b-like planet is $41/422\approx 9.72 \%$ in our ejection model.

Zoom In Zoom Out Reset image size

As known, PPS scenarios can shed light on several observational features of exoplanets, such as the formation of hot Jupiters (Rasio & Ford 1996; Nagasawa & Ida 2011; Beaugé & Nesvorný 2012), the stellar obliquity distribution of stars with hot Jupiters (Winn et al. 2015), and the eccentricity distribution of giant planets (Chatterjee et al. 2008; Jurić & Tremaine 2008). Furthermore, recent studies have shown that some circumbinary planets are likely to have undergone a scattering process. In a single-stellar system, the scattering model can be used to reproduce the eccentricity distribution of extrasolar giant planets. The mass ratio of the planets is a vital parameter to understanding their dynamical evolution.

In the present work, we concentrate on investigating the role of the mass ratio of the circumbinary planets over the scattering results. We first assume that the currently observed single-planet system is the survivor of PPS from an original multi-planetary system. Next, we extensively explore the effect of the mass ratio on the ejection preference and the orbital distribution of the surviving planets. Our simulations showed that the binary is involved in the scattering scenario, which makes the scattering results greatly different from those of PPS in single-star systems. In addition, combined with the disk-driven models of circumbinary planets, we have studied the orbital configuration formation of Kepler-34b and Kepler-413b based on a planet–planet ejection scenario. Thus, we summarize the major conclusions of this work as follows.

1. Ejection preference is related to the planetary mass ratio and the scattering location. If the mass ratio of the two planets is greater than a critical value (~0.3 in our model), the inner planet has an equivalent or even larger probability to be ejected out of the system, as the PPS takes place nearby the unstable boundary of the binary. If the mass ratio is less than the critical value or the scattering position is moved away from the binary, the ejections are always of the less massive planets.

2. The eccentricity distribution of the surviving planets varies with the mass ratio and the initial relative position of the two planets. To obtain a low eccentricity, the surviving planet has to be the more massive one, regardless of its initial location (inner or outer). If the mass ratio of the planets is less than ~0.3, the remaining planets can maintain a small eccentricity, which is nearly equal to their forced eccentricity.

3. Within the range of [1.2–1.8] a_c, the SMA of the surviving planets always increase after PPS. If the innermost region that a planet can reach (driven by the disk) is beyond its currently observed location, the two-planet ejection model cannot reproduce their current orbital configuration.

4. In the migration and ejection scenario, the formation of the configuration of Kepler-34b or Kepler-413b seems to be likely from our simulations. It requires that the planets previously migrated closer to the binary, as indicated in Pierens & Nelson (2013). Its universality needs a more mature disk model to elucidate this in the future.

Compared to the protoplanetary disk in a single-star system, the architecture of the circumbinary disk seems to be more complicated (Fleming & Quinn 2017). The evolution of the disk is relevant to the mass ratio of the binary, its eccentricity, and the physical parameters of the circumbinary disk. Thun et al. (2017) studied how the above factors affect the gap size of the disk. Interestingly, they found that there is a bifurcation occurring at around ${e}_{B}\approx 0.18$ where the gap is smallest. For values of e_B smaller and larger than 0.18, the gap size can increase. It is worth further investigating how this feature of circumbinary disks plays a role in the orbital evolution of a formed planet. In the meantime, planetary accretion, growth, and migration should be considered in the context of the physical evolution of the disk. In particular, the final location of a circumbinary planet is determined by a delicate interplay between such parameters as the detailed structure of the tidal-formed cavity and the orbital parameters of the planet, the dissipation of the disk, the disc self-gravity, etc. Truthfully, these are open questions for the planetary community. However, the disk-driven migration of circumbinary planets and the subsequent PPS make it possible to shape the final orbital configuration of Kepler circumbinary planets, and more detailed scenarios should be investigated in the forthcoming study.

We thank the anonymous referee for constructive comments and suggestions that improved the manuscript. This work is financially supported by National Natural Science Foundation of China (grants No. 11773081, 11573018, 11473073, 11661161013), the Strategic Priority Research Program-The Emergence of Cosmological Structures of the Chinese Academy of Sciences (grant No. XDB09000000), the innovative and interdisciplinary program by CAS (grant No. KJZD-EW-Z001), the Natural Science Foundation of Jiangsu Province (grant No. BK20141509), and the Foundation of Minor Planets of Purple Mountain Observatory. G.Y.-X. also acknowledges the support from Shandong Provincial Natural Science Foundation, China (ZR2014JL004).