Predicting the Configuration of Planetary System: KOI-152 Observed by Kepler
The recent Kepler discovery of KOI-152 reveals a system of three hot super-Earth candidates that are in or near a 4:2:1 mean motion resonance. It is unlikely that they formed in situ, the planets probably underwent orbital migration during the formation and evolution process. The small semimajor axes of the three planets suggest that migration stopped at the inner edge of the primordial gas disk. In this paper we focus on the influence of migration halting mechanisms, including migration ”dead zones”, and inner truncation by the stellar magnetic field. We show that the stellar accretion rate, stellar magnetic field and the speed of migration in the proto-planetary disk are the main factors affecting the final configuration of KOI-152. Our simulations suggest that three planets may be around a star with low star accretion rate or with high magnetic field. On the other hand, slow type I migration, which decreases to one tenth of the linear analysis results, favors forming the configuration of KOI-152. Under such formation scenario, the planets in the system are not massive enough to open gaps in the gas disk. The upper limit of the planetary masses are estimated to be about , and , respectively. Our results are also indicative of the near Laplacian configurations that are quite common in planetary systems.
Subject headings:(stars: KIC 8394721) planetary systems: formation-solar system: formation-stars: individual (KOI-152)
The Kepler Mission, launched in March 2009, photometrically monitors a large patch of sky with sufficient precision to detect terrestrial sized planets in potentially habitable orbits. The Kepler Mission has released their first 16 months data of 2321 transiting planet candidates (Batalha et al., 2012; Fabrycky et al., 2012). The mission is sensitive to a larger range of semimajor axes than ground-based transit surveys (Borucki et al., 2010). Thus there are opportunities to detect multiplanetary systems. According to statistical results on the first four months data, 17% systems have multiple planet candidates (Borucki et al., 2011). More than a dozen multiple transiting planet systems will be confirmed or rejected by means of transit timing variations (TTVs) (Ford et al., 2011). The candidate multiple planetary systems show that at least 16% contain a pair of planets close to 2:1 period commensurability (with a period ratio of two planets ranging from 1.83 to 2.18) (Lissauer et al., 2011). Furthermore, the detailed statistical analysis of the release data over 16 months suggests that the fraction of multiple planetary systems has raised from 17% to 20% and the number of the resonant systems also increases (Batalha et al., 2012). The resonant systems are of value to the researchers, who investigate formation and evolution of the planetary systems. Steffen et al. (2010) analyzed five Kepler target stars and their planet candidates, in which KOI-152 (KOI: Kepler Objects of Interest) is a known system, consisting of three planetary candidates. Moreover, three planets, orbiting about an F dwarf star (KIC 8394721) with a mass of , are very close to 4:2:1 mean motion resonance (MMR) (Steffen et al., 2010). Table 1 shows the orbital elements of the planets in some detail, i.e., 152.01, 152.02, and 152.03 in the order that the transit detection software identified them in the Kepler data, and hereafter we label them as Planet 01, 02, and 03, respectively, for brevity. Their masses are now estimated to be in the range of (20-100) , (9-30) , and (9-30) , respectively. Considering the density limit and formation scenarios, observers may assume them to be 60 , 15 , and 15 , respectively. Although the semimajor axis of Planet 01 is not well determined, estimated value indicates that they are very close to the central star. The orbital period ratios of each pairs are and . The eccentricities are estimated to be zero and no evidence for large eccentricities has been revealed (Steffen et al., 2010). Therefore, we assume the eccentricities to be zero in this work. Based on the possibility of observing such a three-planet system, the planets likely occupy nearly coplanar orbits with a small deviation of inclination from fundamental framework (Steffen et al., 2010).
It is well-known that the planetary configurations involved in 4:2:1 MMR, also occur both in our solar system and exoplanetary systems. For instance, the Galilean moons of Jupiter had been revealed in a three-body Laplace resonance over several hundred years. In addition, another example is that three super-Earths constitute the HD 40307 system, where three planets are near 4:2:1 MMRs (Papaloizou & Terquem, 2010), similar to KOI-152. For HD 40307, three proto-planets formed in the proto-planetary disk with configuration near MMRs, which strongly constrains the planetary formation and orbital migration theories. In this sense, the formation scenario of such near Laplacian configuration is of great interest to the researchers, and such investigations may test some planetary formation theory.
According to core-accretion model, a planet with semimajor axis can grow up with the material surrounding it to mass (Ida & Lin, 2004)
where is the Hill radius of a planet with a mass , and is the so-called feeding zone of the planet, is the mass of central star, is the enhancement factor to the Minimum Mass Solar Nebula (MMSN), is the volatile enhancement for exterior or interior to the snow line with a value of 4.2 or 1, respectively. For KOI-152, we adopt AU (Ida & Lin, 2004). If three planets formed in the region as shown in Table 1 with their estimated minimum masses, should be at least 37. Hence, we may conclude that it is impossible that all planets formed in situ.
Now, it is widely believed that there are mainly two formation mechanisms to produce short-period planets, e.g., planet-planet scattering and planetary migration (Rasio & Ford, 1996). The planet-planet scattering scenario always requires a massive planet to stir up the eccentricities of other bodies and trigger the scattering process. After this process had done, some planets may obtain high eccentricity (Rasio & Ford, 1996), away from their initial locations. However, the planets of KOI-152 are near 4:2:1 MMR with nearly circular orbits, and in this sense it seems to be impossible to tune scattering scenario to yield such a precise configuration.
Another mechanism is that the orbital migration occurs in the gaseous disk (Goldreich & Tremaine, 1980; Lin et al., 1996). Multiple planets are likely to be captured into MMRs given an appropriate migration speed (Masset & Snellgrove, 2001). As MMR commonly emerges in the planetary systems, the orbital migration is now considered as one of the plausible mechanisms to form such systems. Take the GJ 876 system as an example, consisting of four planets, GJ 876 b and GJ 876 c are locked into 2:1 MMR, which can be explained by migration scenario (Lee & Peale, 2002; Ji et al., 2002, 2003; Zhou et al., 2005; Zhang et al., 2010).
Recently, an alternative scenario of collision-merger has been proposed to account for the formation of short-period planets (Ji et al., 2011). In this mechanism, several embryos can be excited by giant planets after the gas of the disk depletes, and then merge into a larger body moving on a close-in orbit. Nevertheless, in these scenarios, planets are not easy to form in MMRs.
Considering the aforementioned factors, we propose a scenario to produce a configuration of KOI-152. First, three planets are assumed to have formed in the region away from the star with their nominal masses. Then, the planets may undergo type I or II migration due to interaction with the gaseous disk until they halt at the inner region of the disk. In this phase, three planets are trapped into MMRs during the migration. Finally, tidal effects, arising from the central star, circularizes their orbits.
Such a scenario had been supposed by Terquem & Papaloizou (2007). They calculated a series of runs that are composed of 10-25 planets or planetary cores in a disk with masses ranging from 0.1 to 1 , which undergo type I migration. They showed that hot super-Earths or Neptunes do not become isolated during their inward migration, and the companions on near-commensurate orbits always survive. Nevertheless, in their work, the authors did not consider the effect of different speed of type I migration, which can occur. In a more recent work, Papaloizou & Terquem (2010) investigated the formation of the HD 40307 system, bearing a resemblance to KOI-152. They adopted a similar formation scenario and found that in the end of the simulations, the planets are driven out of Laplace resonances due to tidal effects with the central star, and they finally reach a planetary configuration very close to HD 40307. Furthermore, Wang & Zhou (2011) accounted for the contribution of type I migration and mainly focused on the speed of migration under the perturbation of gas-giants in the outer region in M dwarf system. They further point out that pairs of planets favor forming not only near the 2:1 MMR but other first order resonances.
One of the important factors that influence final configurations is holes in the gas disk. In general, holes can appear at two positions. One is the boundary of the dead zone and the active zone in the midplane of the disk responding to the magnetorotational instability (MRI). For protostellar gas disk in ad hoc -prescription (Shakura & Sunyaev, 1973), the mass accretion rate, , is constant across the whole disk, where is the disk scale height, , , and are the sound speed of the midplane, the Kepler angular velocity, and the gas density, respectively. As the value of decreases from the active zone ( 0.018; Sato et al. 2000) to the dead zone ( 0.006), in the midplane the value of the gas density in the dead zone (outer) is three times of that in the active zone (inner). Then a maximum density location occurs. Another is the inner hole of the disk caused by the coupling of the star’s magnetic field with the gas. Gas falls onto the surface of the central star under the effect of the torque induced by the stellar magnetic field, a truncation happens at the inner region and an inner hole appears. A maximum density of the gas disk appears at the boundary of the inner hole. Due to the variation of density, the speed of type I migration may be changed. The two mentioned regions play a significant role in forming the final planetary configuration, especially for low-mass planetary systems.
In this work, we focus on exploring the configuration formation of KOI-152, mainly on the following aspects: (a) the speed of type I migration of planets, (b) the density profile of gas disk, (c) the possible range of masses of three planets, and (d) the nature of the star in the system. This paper is organized as follows: In §2, we introduce the adopted disk model, orbital migration and eccentricity damping models in the investigation. In §3., we present the simulation method and outcomes. Finally, we conclude and summarize the results in §4.
2.1. Disk Model
In order to explore the configuration formation, we consider a system consisting of a central star and three planets, which formed far away from the star. We assume that three planets are initially embedded in a gaseous disk. The surface density is given as (Pringle, 1981)
where is the accretion rate of the star and is the effective viscosity at the orbit of a semimajor axis a. According to the observation of young cluster -Oph, the accretion rate of the star can be written as (Natta et al., 2006; Vorobyov & Basu, 2009)
According to equation (3), the accretion rate of star is for this system. Nevertheless, the value will decrease, on average, with the evolution of T Tauri star and its disk. Hence, herein we consider the star accretion rate in the range of [, ] . The effective viscosity of the disk is , where and , represent the efficiency factor of angular momentum transport, and sound speed at the midplane, respectively; means the isothermal density scale height, refers to the Kepler angular velocity (Shakura & Sunyaev, 1973). Because of the effect of MRI, the values of in the active zone and the dead zone are quite different. The effective value of is expressed as (Kretke & Lin, 2007; Kretke et al., 2009)
where , denote the value of in active zone and dead zone, respectively. Herein, we choose and (Sano et al., 2000). The parameter in the error function erf is the location of the boundary of MRI and represents the width of the transition zone. is modeled as (Kretke et al., 2009)
where is the grain opacity.
Considering the disk depletion, the gas density profile can be modified to be
where refers to the disk depletion timescale, which is observed as few million years (Haisch et al, 2001). We adopt yr in the simulations, where is the time of evolution.
Because of the stellar magnetic field, the gas disk is truncated at (Koenigl, 1991)
where , , and refer to the radius of the star, the radius of the sun and the magnetic field of the central star, respectively. is a free parameter. Herein, we choose = 1 corresponding to a typical Alfvn radius in the way of spherical accretion. Hereafter we use and , the Keplerian orbital period, instead of and for convenience. Combining the effect of the magnetic field, the gas density profile is substituted by
is induced by the truncation of the magnetic field.
Based on equation (8), we find that and there are generally two maxima in the gas density profile. Figure 1 shows the density profile with various star accretion rates (see the top panel (a)). We label the inner maximum density location as DM1, similarly, the outer one as DM2. The locations of DM1 and DM2 change with the star accretion rate. In Figure 2 we show the density versus star accretion rate. From Figure 2 (the top panel (a)), we notice that, with a decrease of star accretion rate, the values of DM1 and DM2 approach each other until they merge into one, the combination occurs at , corresponding to a maximum density at an orbital period of 13 days. Herein we choose , , and , respectively, where the bottom panel (b) displays the values of the density at DM1 and DM2, respectively.
2.2. Planetary migration and eccentricity damping
For the planets in KOI-152 system, their masses are estimated to be less than 100 . From a classical planetary formation theory, they may undergo type I or type II migration during their evolution.
Type I migration is induced by angular-momentum exchange between gas disk and planets. Based on linear analysis, the net momentum loss on a planet causes an inward migration (Goldreich & Tremaine, 1979; Ward, 1997; Tanaka et al., 2002). Under this assumption, the speed of type I migration is very fast. In such situation, it is difficult to produce terrestrial planets. Recently, there are several new theories on reducing the speed of type I migration or even reverse the migrating direction (Baruteau & Masset, 2008; Kley & Crida, 2008; Kley et al., 2009; Paardekooper & Papaloizou, 2008; Wang & Zhou, 2011).
Considering the uncertainty of type I migration, we adopt a reduction of the migration speed, taking a timescale of type I migration of an embryo with mass as (Tanaka et al., 2002)
where , and are eccentricity, distance from the central star, scale hight of the disk, and the Keplerian angular velocity, respectively. is the timescale of linear analysis result, is the reduction factor. Herein, we choose = 0.03, 0.1, and 0.3, respectively, and . means the gas density profile of the disk expressed in equation (8). As the value of is related to the density profile, the speed of type I migration may be slowed down or even reversed at some special areas. In addition, Figure 1 shows the values of using = as an example (see the bottom panel (b)). According to equation (10), if , the timescale of type I migration will transfer from positive to negative. Furthermore, we notice that when embryos run across DM2 and DM1, the migration speed will decrease, which may lead to the trapping of the embryos there.
When it grows massive enough, a planet will start to experience type II migration, as the strong torque caused by the planet will open a gap in the gaseous disk (Lin & Papaloizou, 1993). The timescale of type II migration for a planet with mass is (Ida & Lin, 2008)
where is the efficiency factor of angular momentum transport. When the mass of planet is comparable to that of the gas disk, a fraction () of total angular momentum will transfer between the planet and the disk in the evolution. In this case, applies. Herein we adopt , . If the mass of the gas disk is larger than that of the planet, it will migrate with the gas disk over the timescale of . We emphasize that the timescale of type II migration is larger than that of type I.
A gap will form in the gas disk when the planet grows to massive enough () (Ida & Lin, 2008),
In this work, we assume that the planet will undergo type II migration when its mass is greater than .
Additionally, we consider the eccentricity damping induced by the interactions between the gas disk and embryo (Goldreich & Tremaine, 1979). The damping timescale for an embryo with mass is described as (Cresswell & Nelson, 2006)
where is a normalization factor to be consistent with hydrodynamical simulations. The meanings of other symbols are similar to those in equation (10).
3. Numerical Simulations and Results
To explore the secular evolution of the KOI-152 planetary system, we assume that three planets had formed in the outer region of the system and migrate toward the inner region due to the interactions with the gas disk. Thus, the acceleration of a planet with mass is given as
where and mean the position and velocity vectors of the planet in the stellar-centric coordinates, and the external forces are defined as
Each planet is assumed to be initially in coplanar and near-circular orbit and suffers from mutual gravitational interaction and the effect exerted by the central star. The initial orbital elements of each planet were randomly generated: argument of pericentre, longitude of the ascending node, and mean anomaly were randomly set between to . In this way, we generate a set of 34 runs for simulation.
We numerically integrate the equations (14) using a time-symmetric Hermit scheme (Aarseth, 2003). If the distance from a planet to the central star is smaller than the radii of the star we originally set, we assume that the planet collides with the star. Each run evolved for yr.
Based on equation (12), for , if AU, then we have . Hence, if their initial locations are all without 0.125 AU, KOI-152.02 and KOI-152.03 will undergo type I migration, the outmost one will experience type I or II migration. In this sense, two kinds of models are assumed in the simulations: For Model 1, we suppose that all planets will suffer type I migration; whereas for Model 2, we simply consider that KOI-152.02 and KOI-152.03 will go through type I migration but KOI-152.01 will undergo type II migration. According to the aforementioned analysis, one of the planets in this system may be captured at DM1 or DM2. In such case, taking into account the range of the star accretion rate [, ] , we find that the positions of DM2 are at 13, 26, and 48 days, respectively, in three groups for Model 1. In addition, the conditions of density profile of various groups and models are summarized as follows.
Group 1: For , the density profile is labeled as the blue line in Figure 1. DM2 = 48.039 days, KOI-152.01 is likely to be captured at .
Group 2: For , is labeled as the black line (Figure 1). DM2 = 26.08 days, KOI-152.02 is likely to be captured at .
Group 3: For , is labeled as the green line (Figure 1). DM2 = 13.067 days, KOI-152.03 is likely to be captured at .
Group 4: For , is labeled as the red line (Figure 1). DM2 = 51.9 days, KOI-152.01 is likely to be captured at .
Comparison with Group 2 of Model 1: For , labeled as the black line in Figure 1. DM2 = 26.08 days, KOI-152.02 is likely to be captured at . However, KOI-152.01 will undergo type II migration.
|ID||Initial periods||Terminal periods||Terminal periods ratios||,|
|Group 1||120, 320, 850||0.1||2.58, 5.2, 60.5||11.63, 2.02||48.039, 2.443|
|Group 2-1||120, 320, 850||0.03||3.8, 7.75, 32.83||4.24, 2.04||26.08, 3.618|
|Group 2-2||120, 320, 850||0.1||1.899, 12.19, 24||1.97, 6.42||26.08, 3.618|
|Group 2-3||120, 320, 850||0.3||3.8, 7.67, 32.85||4.28, 2.02||26.08, 3.618|
|Group 2-4||120, 320, 20012||0.03||31.72, 64.23, 6960.77||108.37,2.02||26.08, 3.618|
|Group 2-5||120, 320, 20012||0.1||3.82, 7.7, 32.85||4.27, 2.01||26.08, 3.618|
|Group 2-6||120, 320, 20012||0.3||3.78, 5.72, 32.83||5.74, 1.51||26.08, 3.618|
|Group 3-1||120, 320, 850||0.1||3.5, 7.12, 14.73||2.07, 2.03||13.067|
|Group 3-2||70, 150, 320||0.03||13.78, 27.87, 57.07||2.05, 2.02||13.067|
|Group 3-3||220, 500, 1400||0.1||11.71, 23.56, 47.64||2.02, 2.01||13.067|
|Group 4||220, 320, 850||0.1||14.17, 28.56, 58.54||2.05, 2.01||51.9|
|KOI-152||13.48, 27.40, 52.09||1.90, 2.03|
3.1. Model 1: All planets undergo type I migration
In order to understand the influence of the star accretion rate and the speed of type I migration, we perform four groups of simulations. Table 2 are listed the detailed information for the dominatant results of each group, where the last column shows the Keplerian orbital periods at the locations of DM1 and DM2.
Group 1: KOI-152.01 captured at
In this group, DM1 and DM2 occur at 2.4 days and 48 days, separately, with . From the profile of gas density, we note that, if Planet 01 is likely to be trapped at DM2, the resulting configuration may be analogous to KOI-152. In total, we performed five runs to examine how the planets come into the resonant region, where Table 2 reports typical outcomes in our simulations. For example, in a typical run, Planet 02 or 03 will continue to move inward until reaching DM1 in cases where they have been kicked inside of DM2. Subsequently, Planet 01 is trapped at 60 days as shown in Figure 3 and the two inner planets are eventually captured into 2:1 MMR. From Figure 1, we show that if a planet is pushed into the region inside DM2, the value of changes from negative to positive before it approaches DM1. The change in sign of is the reason that the two inner bodies migrate towards DM1. However, no matter how we varied the speed of type I migration or re-scaled the initial position of Planet 01, (e.g., much more distant than the value given in Table 2), Planet 02 and 03 still rush into the inner zone about DM1 in the evolution. Hence, we may safely conclude that a configuration like KOI-152 cannot be generated when the star is too young with high star accretion rate. To sum up the above outcomes, for five runs, we find that Planet 02 and 03 are locked into 2:1 MMR in three cases during dynamical evolution, whereas they are captured into 3:2 MMR in the other two runs.
In this scenario, a planetary configuration is finally created, that consists of two planets trapped into a 2:1 MMR or 3:2 MMR at DM1 and the outmost one resided inside DM2, far away from DM1.
Group 2: KOI-152.02 captured at
By adopting , we have DM1 and DM2 located at days, respectively. In this group, we performed six runs in total. The results of all runs are reported in Table 2. From Table 2, we observe that, similar to the cases of Group 1, Planet 03 is quickly captured at DM1 as it is thrown into the region inside DM2. In Run 1-3, we utilize the same initial conditions but choose a variational migration speed. In these runs, Planet 02 cannot stay at DM2 but is kicked into the inner region by the perturbation of Planet 01, then it falls into MMR with Planet 01 (run 2) or Planet 03 (run 1, 3). In Run 4-6, Planet 01 is initially well separated from the other two, by placing the outer planet in a starting location of much more distant from the central star. The results of Run 5 and 6 are analogous to that of Run 1-3. The variation of is the governing reason for the evolution of Run 1-3 and 5-6. However, owing to slow migration speed, Run 4 differs from other runs, where three planets cannot pass through DM2. Here, one may notice that it is impossible for three planets to form a configuration resembling to KOI-152. From the analysis of six runs, we summarize the simulation outcomes, where Planet 02 and 03 are in a 2:1 MMR for four runs and in one run they are captured into a 3:2 MMR , whereas in another special case Planet 01 and 02 become trapped into a 2:1 MMR over the evolution.
The created configuration in Group 2 varies with the value of . The inner planet is always trapped in the boundary of the inner hole unless they undergo a lower speed of type I migration and the outmost planet is formed farther simultaneously. Thus, in such scenario, a configuration similar to KOI-152 cannot be created.
Group 3: KOI-152.03 captured at
In the simulations, we assume the star accretion rate is equal to . From the density profile shown in Figure 1, we see that DM1 and DM2 tend to combine into one position at 13 days. In this group, we totally carried out 10 runs. Details of three runs are also shown in Table 2. If we examine the type I migration considering , the results of evolution then bear resemblance to the case of Group 3-1, where KOI-152.01 is trapped at 14 days in the meantime the other two planets jump into the inner region at days, unless we started them at distant orbits from the star or slow down the speed of type I migration. In Group 3-2, we lower the migration speed and the result is shown in Figure 4, where Planet 03 is capable to being trapped at 14 days. The simulation results are consistent with the current observational values of KOI-152. When Planet 02 and 01 keep pace with Planet 03, two pairs are all captured into 2:1 MMRs. Group 3-3 simulates the condition related to a high speed of type I migration and distant initial orbits. Figure 5 displays that Planet 01 and 02 are locked into 2:1 MMR in the dynamical evolution as well as Planet 02 and 03. In conclusion, according to our evaluation, three planets are eventually captured into Laplace resonance, with the resonant angles in a libration about 180 for Group 3-2 and 3-3. When the planets are trapped into MMRs, their eccentricities are excited during the evolution; but due to strong gas damping, they cannot be pumped up to large values.
Taking into account, a configuration similar to KOI-152 is formed, where two pairs of planets are in the 2:1 MMRs.
Group 4: High magnetic field star
Herein we consider that the star bears a high magnetic field of KG. In comparison with Group 3, for a high magnetic field star, the truncation of the inner hole is at 50 days, farther than that a low magnetic field star as shown in Figure 6. Herein we carried out three runs. Figure 7 illustrates the results of a typical run, where the middle and bottom panels (c-f) show the resonant angles for the planets. From the figure, we note that three planets are captured into a 4:2:1 MMR at 1 Myr. Simultaneously, we examine the resonant angles and find that three planets are also locked into a Laplacian resonance, librating about 0 at 4 Myr. When they are captured into MMR, Planet 02 and 03 may migrate towards the inner region of the hole with little gas. Consequently, the influence of the gas damping is not dominant. Thus, the eccentricities of Planet 02 and 03 are stirred up to 0.2 when they are trapped into a 2:1 MMR. However, Planet 01 remains outside the hole, and its eccentricity finally decreases to 0.002. For one paradigm, because of fast speed of type I migration, the system is totally destroyed. In another run, the produced planetary configuration is quite analogous to the typical Laplacian geometry, with the resonant angles librating at 180. For , we have a planetary configuration similar to KOI-152.
3.2. Model 2: KOI-152.01 undergoes type II migration
According to our above analysis, Planet 01 may perform type II migration. In this model, we set Planet 02 and 03 to undergo type I migration but Planet 01 suffers from type II migration. We carry out 10 runs for Model 2, by varying the initial position of the outmost planet. But owing to a slow speed of type II migration, Planet 01 cannot reach its nominal position even if it is placed at the orbit with a period of 70 days in the beginning. Figure 8 shows a typical run of the simulations. In this run, the density profile is the same with that in model 1 Group 2, =0.1 for Planet 02 and 03. In the end, the simulation results show that three planets migrate to 31.72, 63.86 and 211.13 days, respectively. However, Planet 01 is a little deviated from the present-day observation.
Under this circumstance, Planet 01 cannot approach its estimated position regardless the initial location of the outmost planet. In addition, the inner two planets are always trapped into 2:1 MMR at DM2. In a word, we cannot generate a Laplacian configuration for three planets from Model 2.
4. Conclusions and Discussions
In this work, we have extensively investigated the formation of configuration for the KOI-152 system using numerical simulations. We assume three planets formed in a region far away from their current locations. The embedded planets in the gas disk will migrate into the regime much closer to the central star. In order to produce a configuration similar to KOI-152, there are some requirements for the disk and the star. In summary, we reach the following conclusions:
1. If KOI-152.01 survives as predicted, no type II migration happened for three planets in the system.
2. From equation (12), the masses of three planets can be limited to be , and , respectively. According to core-accretion model, a solid core may continue to accrete gas envelop until it grows up into 2-10 (Barnes et al., 2009; Bodenheimer et al., 2000; Hubickyj et al., 2005; Ikoma et al., 2001; Pollack et al., 1996), therefore simply judging from the radii of the planets and the formation scenario, we cannot clearly identify the composition of the planets, and their masses are not well determined. Based on the gap opening criteria, the masses of three planets are evaluated to be (9-15) , (9-19) , and (20-24) , respectively.
3. Low star accretion rate and high magnetic field of the star are propitious to the formation of the configuration of KOI-152. Low star accretion rate may imply that the formation of three planets takes place at the later stage of the star evolution, where DM1 and DM2 are merged into one. From Figure 2, we learn that the boundary is about . In this sense, one requirement for the formation of KOI-152 is .
4. Low speed of type I migration with facilitates the formation of the KOI-152 system. This conclusion agrees with those mentioned by Ida & Lin (2008) and Wang & Zhou (2011).
5. In this work, we had not examined tidal interactions between planets and central star after planetary formation scenario. On the basis of tidal timescale (Mardling & Lin, 2004; Zhou & Lin, 2008), the eccentricities of Planet 01, 02, 03, with their nominal orbital elements will be damped in , , , and years, respectively, where is the tidal dissipation factor and we adopt for them. Meanwhile, the semimajor axes are also decreasing over the timescales of . In that case, the orbits of the planets will become a little closer to the star that those without consideration of tidal effects. Owing to shorter timescales, the inner planets migrate faster, driving all of them out of MMR. Then, the configuration of KOI-152 forms and three planets are in near MMRs. This scenario may be suitable to account for the formation of other systems with planetary configurations like KOI-152.
Among 16 months data of Kepler, 242 target stars host two planet
candidates, 85 with three, 25 with four planets, eight with five
planets and one with six (Fabrycky et al., 2012). We examine all candidates
and find that ten systems may have two pairs of the planets involved
in near 2:1 MMRs, e.g., a near Laplacian resonance configuration. The orbital
periods and their ratios are shown in Table
Marcy et al. (2001) revealed that there were two giant planets involved in a 2:1 MMR orbiting the star GJ 876. However, after the fourth planet was discovered in the GJ 876 system, the previously known 2:1 MMR configuration then becomes a Laplacian resonance configuration, where three planets are trapped into a 4:2:1 MMR with the resonant angles librating at 0, respectively, differing from the Galilean moons of Jupiter in a libration at 180 (Rivera et al., 2010). In such configurations, a planetary system will remain stable at least one billion years. A great many of previous works had investigated the dynamics and stability of the system consisting of three planets at then (Ji et al., 2002, 2003; Zhou et al., 2005; Zhang et al., 2010). From the former results, we learn that the migration scenario may be responsible for the formation of two giants which is locked into 2:1 MMR. Thus under the same formation scenario for KOI-152, three planets may be also trapped into a 4:2:1 MMR, showing a resemblance to those of GJ 876. Hence, we may reach a safe conclusion that the near Laplacian configurations are quite common in the planetary systems as also revealed by Kepler and our work may provide some substantial clues to the formation of such intriguing systems.
- affiliation: Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China; email@example.com, firstname.lastname@example.org.
- affiliation: Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China; email@example.com, firstname.lastname@example.org.
- affiliation: Department of Astronomy & Key Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Nanjing University, Nanjing 210093, China.
- Aarseth, S. J. 2003, Gravitational N-Body Simulations, by Sverre J. Aarseth, pp. 430. ISBN 0521432723. Cambridge, UK: Cambridge University Press, November 2003.
- Barnes, R., Jackson, B., Raymond, S. N., West, A. A., & Greenberg, R. 2009, ApJ, 695, 1006
- Baruteau, C., & Masset, F. 2008, ApJ, 672, 1054
- Batalha, N. M., Rowe, J. F., Bryson, S. T., et al. 2012, arXiv:1202.5852
- Bodenheimer, P., Hubickyj, O., & Lissauer, J. J. 2000, Icarus, 143, 2
- Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977
- Borucki, W. J., Koch, D. G., Basri, G., et al. 2011, ApJ, 736, 19
- Cresswell, P., & Nelson, R. P. 2006, A&A, 450, 833
- Fabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. 2012, arXiv:1202.6328
- Ford, E. B., Rowe, J. F., Fabrycky, D. C., et al. 2011, ApJS, 197, 2
- Goldreich, P., & Tremaine, S. 1979, ApJ, 233, 857
- Goldreich, P., & Tremaine, S. 1980, ApJ, 241, 425
- Haisch, K. E., Jr., Lada, E. A., & Lada, C. J. 2001, ApJ, 553, L153
- Hubickyj, O., Bodenheimer, P., & Lissauer, J. J. 2005, Icarus, 179, 415
- Ida, S., & Lin, D. N. C. 2004, ApJ, 604, 388
- Ida, S., & Lin, D. N. C. 2008, ApJ, 673, 487
- Ikoma, M., Emori, H., & Nakazawa, K. 2001, ApJ, 553, 999
- Ji, J., Li, G., & Liu, L. 2002, ApJ, 572, 1041
- Ji, J., Liu, L., Kinoshita, H., et al. 2003, ApJ, 591, L57
- Ji, J., Jin, S., & Tinney, C. G. 2011, ApJ, 727, L5
- Koenigl, A. 1991, ApJ, 370, L39
- Kley, W., & Crida, A. 2008, A&A, 487, L9
- Kley, W., Bitsch, B., & Klahr, H. 2009, A&A, 506, 971
- Kokubo, E., & Ida, S. 2002, ApJ, 581, 666
- Kretke, K. A., & Lin, D. N. C. 2007, ApJ, 664, L55
- Kretke, K. A., Lin, D. N. C., Garaud, P., & Turner, N. J. 2009, ApJ, 690, 407
- Lee M. H., & Peale S. J. 2002, ApJ, 567, 596
- Lin, D. N. C, & Papaloizou, J. C. B. 1993, in: E.H. Levy & J.I. Lunine (eds.), Protostars and Planets III, (Tucson: Unv. Arizona)
- Lin, D. N. C., Bodenheimer, P., & Richardson, D. C. 1996, Nature, 380, 606
- Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, ApJS, 197, 8
- Masset, F., & Snellgrove, M. 2001, MNRAS, 320, L55
- Marcy, G. W., Butler, R. P., Fischer, D., et al. 2001, ApJ, 556, 296
- Mardling, R. A., & Lin, D. N. C. 2004, ApJ, 614, 955
- Natta, A., Testi, L., & Randich, S. 2006, A&A, 452, 245
- Paardekooper, S.-J., & Papaloizou, J. C. B. 2008, A&A, 485, 877
- Papaloizou, J. C. B., & Terquem, C. 2010, MNRAS, 405, 573
- Pollack, J. B.,Hubickyj, O., Bodenheimer, P., et al. 1996, Icarus, 124, 62
- Pringle, J. E. 1981, ARA&A, 19, 137
- Rasio, F. A., & Ford, E. B. 1996, Science, 274, 954
- Rivera, E. J., Laughlin, G., Butler, R. P., et al. 2010, ApJ, 719, 890
- Sano, T., Miyama, S. M., Umebayashi, T., & Nakano, T. 2000, ApJ, 543, 486
- Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337
- Steffen, J. H., Batalha, N. M., Borucki, W. J., et al. 2010, ApJ, 725, 1226
- Tanaka, H., Takeuchi, T., & Ward, W. R. 2002, ApJ, 565, 1257
- Terquem, C., & Papaloizou, J. C. B. 2007, ApJ, 654, 1110
- Vorobyov, E. I., & Basu, S. 2009, ApJ, 703, 922
- Ward, W. R. 1997, Icarus, 126, 261
- Wang, S., & Zhou, J. L. 2011, ApJ, 727, 108
- Zhang, N., Ji, J., & Sun, Z. 2010, MNRAS, 405, 2016
- Zhou, J. L., Aarseth, S. J., Lin, D. N. C., & Nagasawa, M. 2005, ApJ, 631, L85
- Zhou, J. L., & Lin, D. N. C. 2008, IAU Symposium, 249, 285