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Papers by Shigeru Ida

The Astrophysical Journal, Jan 20, 2008

We address two outstanding issues in the sequential accretion scenario for gas giant planet forma... more We address two outstanding issues in the sequential accretion scenario for gas giant planet formation, the retention of dust grains in the presence of gas drag and that of cores despite type I migration. The efficiency of these processes is determined by the disk structure. Theoretical models suggest that planets form in protostellar disk regions with an inactive neutral "dead zone" near the mid plane, sandwiched together by partially ionized surface layers where magnetorotational instability is active. Due to a transition in the abundance of dust grains, the active layer's thickness decreases abruptly near the ice line. Over a range of modest accretion rates (∼ 10 −9 − 10 −8 M ⊙ yr −1), the change in the angular momentum transfer rate leads to local surface density and pressure distribution maxima near the ice line. The azimuthal velocity becomes super-Keplerian and the grains accumulate in this transition zone. This barrier locally retains protoplanetary cores and enhances the heavy element surface density to the critical value needed to initiate efficient gas accretion. It leads to a preferred location and epoch of gas giant formation. We simulate and reproduce the observed frequency and mass-period distribution of gas giants around solar type stars without having to greatly reduce the type I migration strength. The mass function of the short-period planets can be utilized to calibrate the efficiency of type I migration and to extrapolate the fraction of stars with habitable terrestrial planets.

Astronomy and Astrophysics, Jun 1, 2021

Aims. The connection between initial disc conditions and final orbital and physical properties of... more Aims. The connection between initial disc conditions and final orbital and physical properties of planets is not well-understood. In this paper, we numerically study the formation of planetary systems via pebble accretion and investigate the effects of disc properties such as masses, dissipation timescales, and metallicities on planet formation outcomes. Methods. We improved the N-body code SyMBA that was modified for our Paper I by taking account of new planet-disc interaction models and type II migration. We adopted the 'two-α' disc model to mimic the effects of both the standard disc turbulence and the mass accretion driven by the magnetic disc wind. Results. We successfully reproduced the overall distribution trends of semi-major axes, eccentricities, and planetary masses of extrasolar giant planets. There are two types of giant planet formation trends, depending on whether or not the disc's dissipation timescales are comparable to the planet formation timescales. When planet formation happens fast enough, giant planets are fully grown (Jupiter mass or higher) and are distributed widely across the disc. On the other hand, when planet formation is limited by the disc's dissipation, discs generally form low-mass cold Jupiters. Our simulations also naturally explain why hot Jupiters (HJs) tend to be alone and how the observed eccentricity-metallicity trends arise. The low-metallicity discs tend to form nearly circular and coplanar HJs in situ, because planet formation is slower than high-metallicity discs, and thus protoplanetary cores migrate significantly before gas accretion. The high-metallicity discs, on the other hand, generate HJs in situ or via tidal circularisation of eccentric orbits. Both pathways usually involve dynamical instabilities, and thus HJs tend to have broader eccentricity and inclination distributions. When giant planets with very wide orbits ("super-cold Jupiters") are formed via pebble accretion followed by scattering, we predict that they belong to metal-rich stars, have eccentric orbits, and tend to have (∼80%) companions interior to their orbits.

Monthly Notices of the Royal Astronomical Society, Apr 17, 2014

HARPS and Kepler results indicate that half of solar-type stars host planets with periods P < 100... more HARPS and Kepler results indicate that half of solar-type stars host planets with periods P < 100 d and masses M < 30 M ⊕. These super-Earth systems are compact and dynamically cold. Here, we investigate the stability of the super-Earth system around the K-dwarf HD 40307. It could host up to six planets, with one in the habitable zone. We analyse the system's stability using numerical simulations from initial conditions within the observational uncertainties. The most stable solution deviates 3.1σ from the published value, with planets e and f not in resonance and planets b and c apsidally aligned. We study the habitability of the outer planet through the yearly averaged insolation and blackbody temperature at the pole. Both undergo large variations because of its high eccentricity and are much more intense than on Earth. The insolation variations are precession dominated with periods of 40 kyr and 102 kyr for precession and obliquity if the rotation period is 3 d. A rotation period of about 1.5 d could cause extreme obliquity variations because of capture in a Cassini state. For faster rotation rates, the periods converge to 10 and 20 kyr. The large uncertainty in the precession period does not change the overall outcome.

arXiv (Cornell University), Apr 23, 2010

Planet migration is the process by which a planet's orbital radius changes in time. The main agen... more Planet migration is the process by which a planet's orbital radius changes in time. The main agent for causing gas giant planet migration is the gravitational interaction of the young planet with the gaseous disk from which it forms. We describe the migration rates resulting from these interactions based on a simple model for disk properties. These migration rates are higher than is reasonable for planet survival. We discuss some proposed models for which the migration rates are lower. There are major uncertainties in migration rates due to a lack of knowledge about the detailed physical properties of disks. We also describe some additional forms of migration.

Nature Astronomy, 2020

The ice-giant planet Uranus likely underwent a giant impact, given that its spin axis is tilted b... more The ice-giant planet Uranus likely underwent a giant impact, given that its spin axis is tilted by 98 degrees 1-3. That its satellite system is equally inclined and prograde suggests that it was formed as a consequence of the impact. However, the disks predicted by the impact simulations 1, 3, 4 generally have sizes one order smaller and masses two orders larger than those of the observed system at present. Here we show, by means of a theoretical model, that the Uranian satellite formation is regulated by the evolution of the impact-generated disk. Because the vaporization temperature of water ice is low and both Uranus and the impactor are assumed to be ice-dominated, we can conclude that the impact-generated disk has mostly vaporized. We predict that the disk lost a significant amount of water vapour mass and spread to the levels of the current system until the disk cooled down enough for ice condensation and accretion of icy particles to begin. From the predicted distribution of condensed ices, our N-body simulation is able to reproduce the observed mass-orbit configuration of Uranian satellites. This scenario contrasts with the giant-impact model for the Earth's Moon 5 , in which about half of the compact, impact-generated, solid or liquid disk is immediately incorporated into the Moon on impact 6. Uranus has five major satellites in a mass range of 10 −6-10 −4 M U (Fig. 1), where M U 8.7 × 10 25 kg is Uranus mass, extended to ∼ 25 r U , where r U 2.5 × 10 7 m is Uranus' physical radius (Fig. 1). The extension to ∼ 25 r U cannot be accounted for by tidal orbital expansions 7. Their orbits are prograde to Uranus' spin and nearly circular. The total mass of the satellites 1

Monthly Notices of the Royal Astronomical Society, 2019

The detection of Earth-size exoplanets around low-mass stars –in stars such as Proxima Centauri a... more The detection of Earth-size exoplanets around low-mass stars –in stars such as Proxima Centauri and TRAPPIST-1– provide an exceptional chance to improve our understanding of the formation of planets around M stars and brown dwarfs. We explore the formation of such planets with a population synthesis code based on a planetesimal-driven model previously used to study the formation of the Jovian satellites. Because the discs have low mass and the stars are cool, the formation is an inefficient process that happens at short periods, generating compact planetary systems. Planets can be trapped in resonances and we follow the evolution of the planets after the gas has dissipated and they undergo orbit crossings and possible mergers. We find that formation of planets above Mars mass and in the planetesimal accretion scenario, is only possible around stars with masses M⋆ ≥ 0.07Msun and discs of Mdisc ≥ 10−2 Msun. We find that planets above Earth-mass form around stars with masses larger tha...

Astronomy & Astrophysics, 2017

Context. Planet formation with pebbles has been proposed to solve a couple of long-standing issue... more Context. Planet formation with pebbles has been proposed to solve a couple of long-standing issues in the classical formation model. Some sophisticated simulations have been performed to confirm the efficiency of pebble accretion. However, there has not been any global N-body simulations that compare the outcomes of planet formation via pebble accretion to observed extrasolar planetary systems. Aims. In this paper, we study the effects of a range of initial parameters of planet formation via pebble accretion, and present the first results of our simulations. Methods. We incorporate a published pebble-accretion model into the N-body code SyMBA, along with the effects of gas accretion, eccentricity and inclination damping, and planet migration in the disc. Results. We confirm that pebble accretion leads to a variety of planetary systems, but have difficulty in reproducing observed properties of exoplanetary systems, such as planetary mass, semimajor axis, and eccentricity distributions. The main reason behind this is an overly efficient type-I migration, which closely depends on the disc model. However, our simulations also lead to a few interesting predictions. First, we find that formation efficiencies of planets depend on the stellar metallicities, not only for giant planets, but also for Earths (Es) and Super-Earths (SEs). The dependency for Es/SEs is subtle. Although higher metallicity environments lead to faster formation of a larger number of Es/SEs, they also tend to be lost later via dynamical instability. Second, our results indicate that a wide range of bulk densities observed for Es and SEs is a natural consequence of dynamical evolution of planetary systems. Third, the ejection trend of our simulations suggest that one free-floating E/SE may be expected for two smaller-mass planets. Key words. planets and satellites: formation-planets and satellites: dynamical evolution and stability-planetary systemsplanets and satellites: general-protoplanetary disks 1 Recent studies of protostellar discs suggest that the MRI turbulence may not be efficient in the planet-forming region (∼1-10 AU), and that the angular momentum transfer may be largely done by magnetocentrifugal disc winds (Turner et al. 2014). If this were the case, the pressure bumps would need to be created by other mechanisms, or pebbles would need to be trapped by other means such as vortices (e.g. Barge & Sommeria 1995; Raettig et al. 2015).

Progress of Theoretical and Experimental Physics, 2012

In the standard scenario of planet formation, planets are formed from a protoplanetary disk that ... more In the standard scenario of planet formation, planets are formed from a protoplanetary disk that consists of gas and dust. The building blocks of solid planets are called planetesimals; they are formed by coagulation of dust. We review the basic dynamics and accretion of planetesimals by showing N-body simulations. The orbits of planetesimals evolve through two-body gravitational relaxation: viscous stirring increases the random velocity and dynamical friction realizes the equiparation of the random energy. In the early stage of planetesimal accretion, the growth mode of planetesimals is runaway growth, where larger planetesimals grow faster than smaller ones. When a protoplanet (a runaway-growing planetesimal) exceeds a critical mass, the growth mode shifts to oligarchic growth, where similar-sized protoplanets grow while keeping a certain orbital separation. The final stage of terrestrial planet formation is collision among protoplanets, known as giant impacts. We also summarize the dynamical effects of disk gas on planets and the core accretion model for the formation of gas giants, and discuss the diversity of planetary systems.

Astrophys J, 2010

Radial velocity and transit surveys indicate that solar-type stars bear super-Earths, with mass a... more Radial velocity and transit surveys indicate that solar-type stars bear super-Earths, with mass and period up to ∼ 20M ⊕ and a few months, are more common than those with Jupiter-mass gas giants. In many cases, these super-Earths are members of multiple-planet systems in which their mutual dynamical interaction has influenced their formation and evolution. In this paper, we modify an existing numerical population synthesis scheme to take into account protoplanetary embryos' interaction with their evolving natal gaseous disk, as well as their close scatterings and resonant interaction with each other. We show that it is possible for a group of compact embryos to emerge interior to the ice line, grow, migrate, and congregate into closely-packed convoys which stall in the proximity of their host stars. After the disk-gas depletion, they undergo orbit crossing, close scattering, and giant impacts to form multiple rocky Earths or super-Earths in non-resonant orbits around ∼ 0.1AU with moderate eccentricities of ∼ 0.01-0.1. We suggest that most refractory super-Earths with period in the range of a few days to weeks may have formed through this process. These super-Earths differ from Neptune-like ice giants by their compact sizes and lack of a substantial gaseous envelope.

The Astrophysical Journal, 2010

Using orbital integration and analytical arguments, we have found a new mechanism (an "eccentrici... more Using orbital integration and analytical arguments, we have found a new mechanism (an "eccentricity trap") to halt type I migration of planets near the inner edge of a protoplanetary disk. Because asymmetric eccentricity damping due to disk-planet interaction on the innermost planet at the disk edge plays a crucial role in the trap, this mechanism requires continuous eccentricity excitation and hence works for a resonantly interacting convoy of planets. This trap is so strong that the edge torque exerted on the innermost planet can completely halt type I migrations of many outer planets through mutual resonant perturbations. Consequently, the convoy stays outside the disk edge, as a whole. We have derived semi-analytical formula for the condition for the eccentricity trap and predict how many planets are likely to be trapped. We found that several planets or more should be trapped by this mechanism in protoplanetary disks that have cavities. It can be responsible for the formation of non-resonant, multiple, close-in super-Earth systems extending beyond 0.1AU. Such systems are being revealed by radial velocity observations to be quite common around solar-type stars.

Protostars and Planets VI, 2014

With the increasing number of exoplanets discovered, statistical properties of the population as ... more With the increasing number of exoplanets discovered, statistical properties of the population as a whole become unique constraints on planet formation models provided a link between the description of the detailed processes playing a role in this formation and the observed population can be established. Planet population synthesis provides such a link. The approach allows to study how different physical models of individual processes (e.g., proto-planetary disc structure and evolution, planetesimal formation, gas accretion, migration, etc.) affect the overall properties of the population of emerging planets. By necessity, planet population synthesis relies on simplified descriptions of complex processes. These descriptions can be obtained from more detailed specialised simulations of these processes. The objective of this chapter is twofold: 1) provide an overview of the physics entering in the two main approaches to planet population synthesis and 2) present some of the results achieved as well as illustrate how it can be used to extract constraints on the models and to help interpret observations.

We consider several processes operating during the late stages of planet formation that can affec... more We consider several processes operating during the late stages of planet formation that can affect observed orbital elements. Disk-planet interactions, tidal interactions with the central star, long term orbital instability and the Kozai mechanism are discussed.

The Astrophysical Journal, 2003

Most extrasolar planets are observed to have eccentricities much larger than those in the solar s... more Most extrasolar planets are observed to have eccentricities much larger than those in the solar system. Some of these planets have sibling planets, with comparable masses, orbiting around the same host stars. In these multiple planetary systems, eccentricity is modulated by the planets' mutual secular interaction as a consequence of angular momentum exchange between them. For mature planets, the eigenfrequencies of this modulation are determined by their mass and semimajor axis ratios. However, prior to the disk depletion, self-gravity of the planets' nascent disks dominates the precession eigenfrequencies. We examine here the initial evolution of young planets' eccentricity due to the apsidal libration or circulation induced by both the secular interaction between them and the self-gravity of their nascent disks. We show that as the latter effect declines adiabatically with disk depletion, the modulation amplitude of the planets' relative phase of periapsis is approximately invariant despite the time-asymmetrical exchange of angular momentum between planets. However, as the young planets' orbits pass through a state of secular resonance, their mean eccentricities undergo systematic quantitative changes. For applications, we analyze the eccentricity evolution of planets around Andromedae and HD 168443 during the epoch of protostellar disk depletion. We find that the disk depletion can change the planets' eccentricity ratio. However, the relatively large amplitude of the planets' eccentricity cannot be excited if all the planets had small initial eccentricities.

The Astrophysical Journal, 2009

We have investigated planetary accretion from planetesimals in terrestrial planet regions inside ... more We have investigated planetary accretion from planetesimals in terrestrial planet regions inside the ice line around M dwarf stars through N-body simulations including tidal interactions with disk gas. Because of low luminosity of M dwarfs, habitable zones (HZs) are located in inner regions (∼ 0.1AU). In the close-in HZ, type-I migration and the orbital decay induced by eccentricity damping are efficient according to the high disk gas density in the small orbital radii. Since the orbital decay is terminated around the disk inner edge and the disk edge is close to the HZ, the protoplanets accumulated near the disk edge affect formation of planets in the HZ. Ice lines are also in relatively inner regions at ∼ 0.3AU. Due to the small orbital radii, icy protoplanets accrete rapidly and undergo type-I migration before disk depletion. The rapid orbital decay, the proximity of the disk inner edge, and large amount of inflow of icy protoplanets are characteristic in planetary accretion in terrestrial planet regions around M dwarfs. In the case of full efficiency of type-I migration predicted by the linear theory, we found that protoplanets that migrate to the vicinity of the host star undergo close scatterings and collisions, and 4 to 6 planets eventually remain in mutual mean motion resonances and their orbits have small eccentricities (0.01) and they are stable both before and after disk gas decays. In the case of slow migration, the resonant capture is so efficient that denselypacked ∼ 40 small protoplanets remain in mutual mean motion resonances. In this case, they start orbit crossing, after the disk gas decays and eccentricity damping due to tidal interaction with gas is no more effective. Through merging of the protoplanets, several planets in widely-separated non-resonant orbits with relatively large eccentricities (∼ 0.05) are formed. Thus, the final orbital configurations (separations, resonant or nonresonant, eccentricity, distribution) of the terrestrial planets around M dwarfs sensitively depend on strength of type-I migration. We also found that large amount of water-ice is delivered by type-I migration from outer regions and final planets near the inner disk edge around M dwarfs are generally abundant in water-ice except for the innermost one that is shielded by the outer planets, unless type-I migration speed is reduced by a factor of more than 100 from that predicted by the linear theory.

The Astrophysical Journal, 2011

Recent radial-velocity surveys for GK clump giants have revealed that planets also exist around ∼... more Recent radial-velocity surveys for GK clump giants have revealed that planets also exist around ∼1.5-3 M ⊙ stars. However, no planets have been found inside 0.6 AU around clump giants, in contrast to solar-type main-sequence stars, many of which harbor short-period planets such as hot Jupiters. In this study we examine the possibility that planets were engulfed by host stars evolving on the red-giant branch (RGB). We integrate the orbital evolution of planets in the RGB and helium burning (HeB) phases of host stars, including the effects of stellar tide and stellar mass loss. Then we derive the critical semimajor axis (or the survival limit) inside which planets are eventually engulfed by their host stars after tidal decay of their orbits. Especially, we investigate the impact of stellar mass and other stellar parameters on the survival limit in more detail than previous studies. In addition, we make detailed comparison with measured semimajor axes of planets detected so far, which no previous study did. We find that the critical semimajor axis is quite sensitive to stellar mass in the range between 1.7 and 2.1 M ⊙ , which suggests a need for careful comparison between theoretical and observational limits of existence of planets. Our comparison demonstrates that all those planets are beyond the survival limit, which is consistent with the planet-engulfment hypothesis. However, on the high-mass side (> 2.1M ⊙), the detected planets are orbiting significantly far from the survival limit, which suggests that engulfment by host stars may not be the main reason for the observed lack of short-period giant planets. To confirm our conclusion, the detection of more planets around clump giants, especially with masses 2.5M ⊙ , is required.

The Astrophysical Journal, 2012

As the number of discovered extrasolar planets has been increasing, diversity of planetary system... more As the number of discovered extrasolar planets has been increasing, diversity of planetary systems requires studies of new formation scenarios. It is important to study satellite formation in circumplanetary disks, which is often viewed as analogous to formation of rocky planets in protoplanetary disks. We investigated satellite formation from satellitesimals around giant planets through N-body simulations that include gravitational interactions with a circumplanetary gas disk. Our main aim is to reproduce the observable properties of the Galilean satellites around Jupiter through numerical simulations, as previous N-body simulations have not explained the origin of the resonant configuration. We performed accretion simulations based on the work of Sasaki et al. (2010), in which an inner cavity is added to the model of Canup & Ward (2002, 2006). We found that several satellites are formed and captured in mutual mean motion resonances outside the disk inner edge and are stable after rapid disk gas dissipation, which explains the characteristics of the Galilean satellites. In addition, owing to the existence of the disk edge, a radial compositional gradient of the Galilean satellites can also be reproduced. An additional objective of this study is to discuss orbital properties of formed satellites for a wide range of conditions by considering large uncertainties in model parameters. Through numerical experiments and semianalytical arguments, we determined that if the inner edge of a disk is introduced, a Galilean-like configuration in which several satellites are captured into a 2:1 resonance outside the disk inner cavity is almost universal. In fact, such a configuration is produced even for a massive disk 10 4 g cm −2 and rapid type I migration. This result implies the inevitability of a Galilean satellite formation in addition to providing theoretical predictions for extrasolar satellites. That is, we can predict a substantial number of exomoon systems in the 2:1 mean motion resonance close to their host planets awaiting discovery.

The Astrophysical Journal, 2008

In a further development of a deterministic planet formation model (Ida & Lin), we consider the e... more In a further development of a deterministic planet formation model (Ida & Lin), we consider the effect of type I migration of protoplanetary embryos due to their tidal interaction with their nascent disks. During the early phase of protostellar disks, although embryos rapidly emerge in regions interior to the ice line, uninhibited type I migration leads to their efficient self-clearing. But embryos continue to form from residual planetesimals, repeatedly migrate inward, and provide a main channel of heavy-element accretion onto their host stars. During the advanced stages of disk evolution (a few Myr), the gas surface density declines to values comparable to or smaller than that of the minimum mass nebula model, and type I migration is no longer effective for Mars-mass embryos. Over wide ranges of initial disk surface densities and type I migration efficiencies, the surviving population of embryos interior to the ice line has a total mass of several M È. With this reservoir, there is an adequate inventory of residual embryos to subsequently assemble into rocky planets similar to those around the Sun. However, the onset of efficient gas accretion requires the emergence and retention of cores more massive than a few M È prior to the severe depletion of the disk gas. The formation probability of gas giant planets and hence the predicted mass and semimajor axis distributions of extrasolar gas giants are sensitively determined by the strength of type I migration. We suggest that the distributions consistent with observations can be reproduced only if the actual type I migration timescale is at least an order of magnitude longer than that deduced from linear theories.

The Astrophysical Journal, 2008

We have investigated the formation of close-in extrasolar giant planets through a coupling effect... more We have investigated the formation of close-in extrasolar giant planets through a coupling effect of mutual scattering, Kozai mechanism, and tidal circularization, by orbital integrations. Close-in gas giants would have been originally formed at several AU's beyond the ice lines in protoplanetary disks and migrated close to their host stars. Although type II migration due to planet-disk interactions may be a major channel for the migration, we show that this scattering process would also give a non-negligible contribution. We have carried out orbital integrations of three planets with Jupiter-mass, directly including the effect of tidal circularization. We have found that in about 30% runs close-in planets are formed, which is much higher than suggested by previous studies. Three-planet orbit crossing usually results in one or two planets ejection. The tidal circularization often occurs during the three-planet orbit crossing, but previous studies have monitored only the final stage after the ejection, significantly underestimating the formation probability. We have found that Kozai mechanism by outer planets is responsible for the formation of close-in planets. During the three-planet orbital crossing, the Kozai excitation is repeated and the eccentricity is often increased secularly to values close enough to unity for tidal circularization to transform the inner planet to a close-in planet. Since a moderate eccentricity can remain for the close-in planet, this mechanism may account for the observed close-in planets with moderate eccentricities and without nearby secondary planets. Since these planets also remain a broad range of orbital inclinations (even retrograde ones), the contribution of this process would be clarified by more observations of Rossiter-McLaughlin effects for transiting planets.

The Astrophysical Journal, 2010

The Jovian regular satellite system mainly consists of four Galilean satellites that have similar... more The Jovian regular satellite system mainly consists of four Galilean satellites that have similar masses and are trapped in mutual mean motion resonances except for the outer satellite, Callisto. On the other hand, the Saturnian regular satellite system has only one big icy body, Titan, and a population of much smaller icy moons. We have investigated the origin of these major differences between the Jovian and Saturnian satellite systems by semi-analytically simulating the growth and orbital migration of proto-satellites in an accreting proto-satellite disk. We set up two different disk evolution/structure models that correspond to Jovian and Saturnian systems, by building upon previously developed models of an actively-supplied proto-satellite disk, the formation of gas giants, and observations of young stars. Our simulations extend previous models by including the (1) different termination timescales of gas infall onto the protosatellite disk and (2) different evolution of a cavity in the disk, between the Jovian and Saturnian systems. We have performed Monte Carlo simulations and show that in the case of the Jovian systems, four to five similar-mass satellites are likely to remain trapped in mean motion resonances. This orbital configuration is formed by type I * a log uniform distribution

Icarus, 2012

Many extrasolar planetary systems containing multiple super-Earths have been discovered. N-body s... more Many extrasolar planetary systems containing multiple super-Earths have been discovered. N-body simulations taking into account standard type-I planetary migration suggest that protoplanets are captured into mean-motion resonant orbits near the inner disk edge at which the migration is halted. Previous N-body simulations suggested that orbital stability of the resonant systems depends on number of the captured planets. In the unstable case, through close scattering and merging between planets, non-resonant multiple systems are finally formed. In this paper, we investigate the critical number of the resonantly trapped planets beyond which orbital instability occurs after disk gas depletion. We find that when the total number of planets (N) is larger than the critical number (N crit), crossing time that is a timescale of initiation of the orbital instability is similar to non-resonant cases, while the orbital instability never occurs within the orbital calculation time (10 8 Kepler time) for N ≤ N crit. Thus, the transition of crossing time across the critical number is drastic. When all the planets are trapped in 7:6 resonance of adjacent pairs, N crit = 4. We examine the dependence of the critical number of 4:3, 6:5 and 8:7 resonance by changing the orbital separation in mutual Hill radii and planetary mass. The critical number increases with increasing the orbital separation in mutual Hill radii with fixed planetary mass and increases with increasing planetary mass with fixed the orbital separation in mutual Hill radii. We also calculate the case of a system which is not composed of

The Astrophysical Journal, Jan 20, 2008

We address two outstanding issues in the sequential accretion scenario for gas giant planet forma... more We address two outstanding issues in the sequential accretion scenario for gas giant planet formation, the retention of dust grains in the presence of gas drag and that of cores despite type I migration. The efficiency of these processes is determined by the disk structure. Theoretical models suggest that planets form in protostellar disk regions with an inactive neutral "dead zone" near the mid plane, sandwiched together by partially ionized surface layers where magnetorotational instability is active. Due to a transition in the abundance of dust grains, the active layer's thickness decreases abruptly near the ice line. Over a range of modest accretion rates (∼ 10 −9 − 10 −8 M ⊙ yr −1), the change in the angular momentum transfer rate leads to local surface density and pressure distribution maxima near the ice line. The azimuthal velocity becomes super-Keplerian and the grains accumulate in this transition zone. This barrier locally retains protoplanetary cores and enhances the heavy element surface density to the critical value needed to initiate efficient gas accretion. It leads to a preferred location and epoch of gas giant formation. We simulate and reproduce the observed frequency and mass-period distribution of gas giants around solar type stars without having to greatly reduce the type I migration strength. The mass function of the short-period planets can be utilized to calibrate the efficiency of type I migration and to extrapolate the fraction of stars with habitable terrestrial planets.

Astronomy and Astrophysics, Jun 1, 2021

Aims. The connection between initial disc conditions and final orbital and physical properties of... more Aims. The connection between initial disc conditions and final orbital and physical properties of planets is not well-understood. In this paper, we numerically study the formation of planetary systems via pebble accretion and investigate the effects of disc properties such as masses, dissipation timescales, and metallicities on planet formation outcomes. Methods. We improved the N-body code SyMBA that was modified for our Paper I by taking account of new planet-disc interaction models and type II migration. We adopted the 'two-α' disc model to mimic the effects of both the standard disc turbulence and the mass accretion driven by the magnetic disc wind. Results. We successfully reproduced the overall distribution trends of semi-major axes, eccentricities, and planetary masses of extrasolar giant planets. There are two types of giant planet formation trends, depending on whether or not the disc's dissipation timescales are comparable to the planet formation timescales. When planet formation happens fast enough, giant planets are fully grown (Jupiter mass or higher) and are distributed widely across the disc. On the other hand, when planet formation is limited by the disc's dissipation, discs generally form low-mass cold Jupiters. Our simulations also naturally explain why hot Jupiters (HJs) tend to be alone and how the observed eccentricity-metallicity trends arise. The low-metallicity discs tend to form nearly circular and coplanar HJs in situ, because planet formation is slower than high-metallicity discs, and thus protoplanetary cores migrate significantly before gas accretion. The high-metallicity discs, on the other hand, generate HJs in situ or via tidal circularisation of eccentric orbits. Both pathways usually involve dynamical instabilities, and thus HJs tend to have broader eccentricity and inclination distributions. When giant planets with very wide orbits ("super-cold Jupiters") are formed via pebble accretion followed by scattering, we predict that they belong to metal-rich stars, have eccentric orbits, and tend to have (∼80%) companions interior to their orbits.

Monthly Notices of the Royal Astronomical Society, Apr 17, 2014

HARPS and Kepler results indicate that half of solar-type stars host planets with periods P < 100... more HARPS and Kepler results indicate that half of solar-type stars host planets with periods P < 100 d and masses M < 30 M ⊕. These super-Earth systems are compact and dynamically cold. Here, we investigate the stability of the super-Earth system around the K-dwarf HD 40307. It could host up to six planets, with one in the habitable zone. We analyse the system's stability using numerical simulations from initial conditions within the observational uncertainties. The most stable solution deviates 3.1σ from the published value, with planets e and f not in resonance and planets b and c apsidally aligned. We study the habitability of the outer planet through the yearly averaged insolation and blackbody temperature at the pole. Both undergo large variations because of its high eccentricity and are much more intense than on Earth. The insolation variations are precession dominated with periods of 40 kyr and 102 kyr for precession and obliquity if the rotation period is 3 d. A rotation period of about 1.5 d could cause extreme obliquity variations because of capture in a Cassini state. For faster rotation rates, the periods converge to 10 and 20 kyr. The large uncertainty in the precession period does not change the overall outcome.

arXiv (Cornell University), Apr 23, 2010

Planet migration is the process by which a planet's orbital radius changes in time. The main agen... more Planet migration is the process by which a planet's orbital radius changes in time. The main agent for causing gas giant planet migration is the gravitational interaction of the young planet with the gaseous disk from which it forms. We describe the migration rates resulting from these interactions based on a simple model for disk properties. These migration rates are higher than is reasonable for planet survival. We discuss some proposed models for which the migration rates are lower. There are major uncertainties in migration rates due to a lack of knowledge about the detailed physical properties of disks. We also describe some additional forms of migration.

Nature Astronomy, 2020

The ice-giant planet Uranus likely underwent a giant impact, given that its spin axis is tilted b... more The ice-giant planet Uranus likely underwent a giant impact, given that its spin axis is tilted by 98 degrees 1-3. That its satellite system is equally inclined and prograde suggests that it was formed as a consequence of the impact. However, the disks predicted by the impact simulations 1, 3, 4 generally have sizes one order smaller and masses two orders larger than those of the observed system at present. Here we show, by means of a theoretical model, that the Uranian satellite formation is regulated by the evolution of the impact-generated disk. Because the vaporization temperature of water ice is low and both Uranus and the impactor are assumed to be ice-dominated, we can conclude that the impact-generated disk has mostly vaporized. We predict that the disk lost a significant amount of water vapour mass and spread to the levels of the current system until the disk cooled down enough for ice condensation and accretion of icy particles to begin. From the predicted distribution of condensed ices, our N-body simulation is able to reproduce the observed mass-orbit configuration of Uranian satellites. This scenario contrasts with the giant-impact model for the Earth's Moon 5 , in which about half of the compact, impact-generated, solid or liquid disk is immediately incorporated into the Moon on impact 6. Uranus has five major satellites in a mass range of 10 −6-10 −4 M U (Fig. 1), where M U 8.7 × 10 25 kg is Uranus mass, extended to ∼ 25 r U , where r U 2.5 × 10 7 m is Uranus' physical radius (Fig. 1). The extension to ∼ 25 r U cannot be accounted for by tidal orbital expansions 7. Their orbits are prograde to Uranus' spin and nearly circular. The total mass of the satellites 1

Monthly Notices of the Royal Astronomical Society, 2019

The detection of Earth-size exoplanets around low-mass stars –in stars such as Proxima Centauri a... more The detection of Earth-size exoplanets around low-mass stars –in stars such as Proxima Centauri and TRAPPIST-1– provide an exceptional chance to improve our understanding of the formation of planets around M stars and brown dwarfs. We explore the formation of such planets with a population synthesis code based on a planetesimal-driven model previously used to study the formation of the Jovian satellites. Because the discs have low mass and the stars are cool, the formation is an inefficient process that happens at short periods, generating compact planetary systems. Planets can be trapped in resonances and we follow the evolution of the planets after the gas has dissipated and they undergo orbit crossings and possible mergers. We find that formation of planets above Mars mass and in the planetesimal accretion scenario, is only possible around stars with masses M⋆ ≥ 0.07Msun and discs of Mdisc ≥ 10−2 Msun. We find that planets above Earth-mass form around stars with masses larger tha...

Astronomy & Astrophysics, 2017

Context. Planet formation with pebbles has been proposed to solve a couple of long-standing issue... more Context. Planet formation with pebbles has been proposed to solve a couple of long-standing issues in the classical formation model. Some sophisticated simulations have been performed to confirm the efficiency of pebble accretion. However, there has not been any global N-body simulations that compare the outcomes of planet formation via pebble accretion to observed extrasolar planetary systems. Aims. In this paper, we study the effects of a range of initial parameters of planet formation via pebble accretion, and present the first results of our simulations. Methods. We incorporate a published pebble-accretion model into the N-body code SyMBA, along with the effects of gas accretion, eccentricity and inclination damping, and planet migration in the disc. Results. We confirm that pebble accretion leads to a variety of planetary systems, but have difficulty in reproducing observed properties of exoplanetary systems, such as planetary mass, semimajor axis, and eccentricity distributions. The main reason behind this is an overly efficient type-I migration, which closely depends on the disc model. However, our simulations also lead to a few interesting predictions. First, we find that formation efficiencies of planets depend on the stellar metallicities, not only for giant planets, but also for Earths (Es) and Super-Earths (SEs). The dependency for Es/SEs is subtle. Although higher metallicity environments lead to faster formation of a larger number of Es/SEs, they also tend to be lost later via dynamical instability. Second, our results indicate that a wide range of bulk densities observed for Es and SEs is a natural consequence of dynamical evolution of planetary systems. Third, the ejection trend of our simulations suggest that one free-floating E/SE may be expected for two smaller-mass planets. Key words. planets and satellites: formation-planets and satellites: dynamical evolution and stability-planetary systemsplanets and satellites: general-protoplanetary disks 1 Recent studies of protostellar discs suggest that the MRI turbulence may not be efficient in the planet-forming region (∼1-10 AU), and that the angular momentum transfer may be largely done by magnetocentrifugal disc winds (Turner et al. 2014). If this were the case, the pressure bumps would need to be created by other mechanisms, or pebbles would need to be trapped by other means such as vortices (e.g. Barge & Sommeria 1995; Raettig et al. 2015).

Progress of Theoretical and Experimental Physics, 2012

In the standard scenario of planet formation, planets are formed from a protoplanetary disk that ... more In the standard scenario of planet formation, planets are formed from a protoplanetary disk that consists of gas and dust. The building blocks of solid planets are called planetesimals; they are formed by coagulation of dust. We review the basic dynamics and accretion of planetesimals by showing N-body simulations. The orbits of planetesimals evolve through two-body gravitational relaxation: viscous stirring increases the random velocity and dynamical friction realizes the equiparation of the random energy. In the early stage of planetesimal accretion, the growth mode of planetesimals is runaway growth, where larger planetesimals grow faster than smaller ones. When a protoplanet (a runaway-growing planetesimal) exceeds a critical mass, the growth mode shifts to oligarchic growth, where similar-sized protoplanets grow while keeping a certain orbital separation. The final stage of terrestrial planet formation is collision among protoplanets, known as giant impacts. We also summarize the dynamical effects of disk gas on planets and the core accretion model for the formation of gas giants, and discuss the diversity of planetary systems.

Astrophys J, 2010

Radial velocity and transit surveys indicate that solar-type stars bear super-Earths, with mass a... more Radial velocity and transit surveys indicate that solar-type stars bear super-Earths, with mass and period up to ∼ 20M ⊕ and a few months, are more common than those with Jupiter-mass gas giants. In many cases, these super-Earths are members of multiple-planet systems in which their mutual dynamical interaction has influenced their formation and evolution. In this paper, we modify an existing numerical population synthesis scheme to take into account protoplanetary embryos' interaction with their evolving natal gaseous disk, as well as their close scatterings and resonant interaction with each other. We show that it is possible for a group of compact embryos to emerge interior to the ice line, grow, migrate, and congregate into closely-packed convoys which stall in the proximity of their host stars. After the disk-gas depletion, they undergo orbit crossing, close scattering, and giant impacts to form multiple rocky Earths or super-Earths in non-resonant orbits around ∼ 0.1AU with moderate eccentricities of ∼ 0.01-0.1. We suggest that most refractory super-Earths with period in the range of a few days to weeks may have formed through this process. These super-Earths differ from Neptune-like ice giants by their compact sizes and lack of a substantial gaseous envelope.

The Astrophysical Journal, 2010

Using orbital integration and analytical arguments, we have found a new mechanism (an "eccentrici... more Using orbital integration and analytical arguments, we have found a new mechanism (an "eccentricity trap") to halt type I migration of planets near the inner edge of a protoplanetary disk. Because asymmetric eccentricity damping due to disk-planet interaction on the innermost planet at the disk edge plays a crucial role in the trap, this mechanism requires continuous eccentricity excitation and hence works for a resonantly interacting convoy of planets. This trap is so strong that the edge torque exerted on the innermost planet can completely halt type I migrations of many outer planets through mutual resonant perturbations. Consequently, the convoy stays outside the disk edge, as a whole. We have derived semi-analytical formula for the condition for the eccentricity trap and predict how many planets are likely to be trapped. We found that several planets or more should be trapped by this mechanism in protoplanetary disks that have cavities. It can be responsible for the formation of non-resonant, multiple, close-in super-Earth systems extending beyond 0.1AU. Such systems are being revealed by radial velocity observations to be quite common around solar-type stars.

Protostars and Planets VI, 2014

With the increasing number of exoplanets discovered, statistical properties of the population as ... more With the increasing number of exoplanets discovered, statistical properties of the population as a whole become unique constraints on planet formation models provided a link between the description of the detailed processes playing a role in this formation and the observed population can be established. Planet population synthesis provides such a link. The approach allows to study how different physical models of individual processes (e.g., proto-planetary disc structure and evolution, planetesimal formation, gas accretion, migration, etc.) affect the overall properties of the population of emerging planets. By necessity, planet population synthesis relies on simplified descriptions of complex processes. These descriptions can be obtained from more detailed specialised simulations of these processes. The objective of this chapter is twofold: 1) provide an overview of the physics entering in the two main approaches to planet population synthesis and 2) present some of the results achieved as well as illustrate how it can be used to extract constraints on the models and to help interpret observations.

We consider several processes operating during the late stages of planet formation that can affec... more We consider several processes operating during the late stages of planet formation that can affect observed orbital elements. Disk-planet interactions, tidal interactions with the central star, long term orbital instability and the Kozai mechanism are discussed.

The Astrophysical Journal, 2003

Most extrasolar planets are observed to have eccentricities much larger than those in the solar s... more Most extrasolar planets are observed to have eccentricities much larger than those in the solar system. Some of these planets have sibling planets, with comparable masses, orbiting around the same host stars. In these multiple planetary systems, eccentricity is modulated by the planets' mutual secular interaction as a consequence of angular momentum exchange between them. For mature planets, the eigenfrequencies of this modulation are determined by their mass and semimajor axis ratios. However, prior to the disk depletion, self-gravity of the planets' nascent disks dominates the precession eigenfrequencies. We examine here the initial evolution of young planets' eccentricity due to the apsidal libration or circulation induced by both the secular interaction between them and the self-gravity of their nascent disks. We show that as the latter effect declines adiabatically with disk depletion, the modulation amplitude of the planets' relative phase of periapsis is approximately invariant despite the time-asymmetrical exchange of angular momentum between planets. However, as the young planets' orbits pass through a state of secular resonance, their mean eccentricities undergo systematic quantitative changes. For applications, we analyze the eccentricity evolution of planets around Andromedae and HD 168443 during the epoch of protostellar disk depletion. We find that the disk depletion can change the planets' eccentricity ratio. However, the relatively large amplitude of the planets' eccentricity cannot be excited if all the planets had small initial eccentricities.

The Astrophysical Journal, 2009

We have investigated planetary accretion from planetesimals in terrestrial planet regions inside ... more We have investigated planetary accretion from planetesimals in terrestrial planet regions inside the ice line around M dwarf stars through N-body simulations including tidal interactions with disk gas. Because of low luminosity of M dwarfs, habitable zones (HZs) are located in inner regions (∼ 0.1AU). In the close-in HZ, type-I migration and the orbital decay induced by eccentricity damping are efficient according to the high disk gas density in the small orbital radii. Since the orbital decay is terminated around the disk inner edge and the disk edge is close to the HZ, the protoplanets accumulated near the disk edge affect formation of planets in the HZ. Ice lines are also in relatively inner regions at ∼ 0.3AU. Due to the small orbital radii, icy protoplanets accrete rapidly and undergo type-I migration before disk depletion. The rapid orbital decay, the proximity of the disk inner edge, and large amount of inflow of icy protoplanets are characteristic in planetary accretion in terrestrial planet regions around M dwarfs. In the case of full efficiency of type-I migration predicted by the linear theory, we found that protoplanets that migrate to the vicinity of the host star undergo close scatterings and collisions, and 4 to 6 planets eventually remain in mutual mean motion resonances and their orbits have small eccentricities (0.01) and they are stable both before and after disk gas decays. In the case of slow migration, the resonant capture is so efficient that denselypacked ∼ 40 small protoplanets remain in mutual mean motion resonances. In this case, they start orbit crossing, after the disk gas decays and eccentricity damping due to tidal interaction with gas is no more effective. Through merging of the protoplanets, several planets in widely-separated non-resonant orbits with relatively large eccentricities (∼ 0.05) are formed. Thus, the final orbital configurations (separations, resonant or nonresonant, eccentricity, distribution) of the terrestrial planets around M dwarfs sensitively depend on strength of type-I migration. We also found that large amount of water-ice is delivered by type-I migration from outer regions and final planets near the inner disk edge around M dwarfs are generally abundant in water-ice except for the innermost one that is shielded by the outer planets, unless type-I migration speed is reduced by a factor of more than 100 from that predicted by the linear theory.

The Astrophysical Journal, 2011

Recent radial-velocity surveys for GK clump giants have revealed that planets also exist around ∼... more Recent radial-velocity surveys for GK clump giants have revealed that planets also exist around ∼1.5-3 M ⊙ stars. However, no planets have been found inside 0.6 AU around clump giants, in contrast to solar-type main-sequence stars, many of which harbor short-period planets such as hot Jupiters. In this study we examine the possibility that planets were engulfed by host stars evolving on the red-giant branch (RGB). We integrate the orbital evolution of planets in the RGB and helium burning (HeB) phases of host stars, including the effects of stellar tide and stellar mass loss. Then we derive the critical semimajor axis (or the survival limit) inside which planets are eventually engulfed by their host stars after tidal decay of their orbits. Especially, we investigate the impact of stellar mass and other stellar parameters on the survival limit in more detail than previous studies. In addition, we make detailed comparison with measured semimajor axes of planets detected so far, which no previous study did. We find that the critical semimajor axis is quite sensitive to stellar mass in the range between 1.7 and 2.1 M ⊙ , which suggests a need for careful comparison between theoretical and observational limits of existence of planets. Our comparison demonstrates that all those planets are beyond the survival limit, which is consistent with the planet-engulfment hypothesis. However, on the high-mass side (> 2.1M ⊙), the detected planets are orbiting significantly far from the survival limit, which suggests that engulfment by host stars may not be the main reason for the observed lack of short-period giant planets. To confirm our conclusion, the detection of more planets around clump giants, especially with masses 2.5M ⊙ , is required.

The Astrophysical Journal, 2012

As the number of discovered extrasolar planets has been increasing, diversity of planetary system... more As the number of discovered extrasolar planets has been increasing, diversity of planetary systems requires studies of new formation scenarios. It is important to study satellite formation in circumplanetary disks, which is often viewed as analogous to formation of rocky planets in protoplanetary disks. We investigated satellite formation from satellitesimals around giant planets through N-body simulations that include gravitational interactions with a circumplanetary gas disk. Our main aim is to reproduce the observable properties of the Galilean satellites around Jupiter through numerical simulations, as previous N-body simulations have not explained the origin of the resonant configuration. We performed accretion simulations based on the work of Sasaki et al. (2010), in which an inner cavity is added to the model of Canup & Ward (2002, 2006). We found that several satellites are formed and captured in mutual mean motion resonances outside the disk inner edge and are stable after rapid disk gas dissipation, which explains the characteristics of the Galilean satellites. In addition, owing to the existence of the disk edge, a radial compositional gradient of the Galilean satellites can also be reproduced. An additional objective of this study is to discuss orbital properties of formed satellites for a wide range of conditions by considering large uncertainties in model parameters. Through numerical experiments and semianalytical arguments, we determined that if the inner edge of a disk is introduced, a Galilean-like configuration in which several satellites are captured into a 2:1 resonance outside the disk inner cavity is almost universal. In fact, such a configuration is produced even for a massive disk 10 4 g cm −2 and rapid type I migration. This result implies the inevitability of a Galilean satellite formation in addition to providing theoretical predictions for extrasolar satellites. That is, we can predict a substantial number of exomoon systems in the 2:1 mean motion resonance close to their host planets awaiting discovery.

The Astrophysical Journal, 2008

In a further development of a deterministic planet formation model (Ida & Lin), we consider the e... more In a further development of a deterministic planet formation model (Ida & Lin), we consider the effect of type I migration of protoplanetary embryos due to their tidal interaction with their nascent disks. During the early phase of protostellar disks, although embryos rapidly emerge in regions interior to the ice line, uninhibited type I migration leads to their efficient self-clearing. But embryos continue to form from residual planetesimals, repeatedly migrate inward, and provide a main channel of heavy-element accretion onto their host stars. During the advanced stages of disk evolution (a few Myr), the gas surface density declines to values comparable to or smaller than that of the minimum mass nebula model, and type I migration is no longer effective for Mars-mass embryos. Over wide ranges of initial disk surface densities and type I migration efficiencies, the surviving population of embryos interior to the ice line has a total mass of several M È. With this reservoir, there is an adequate inventory of residual embryos to subsequently assemble into rocky planets similar to those around the Sun. However, the onset of efficient gas accretion requires the emergence and retention of cores more massive than a few M È prior to the severe depletion of the disk gas. The formation probability of gas giant planets and hence the predicted mass and semimajor axis distributions of extrasolar gas giants are sensitively determined by the strength of type I migration. We suggest that the distributions consistent with observations can be reproduced only if the actual type I migration timescale is at least an order of magnitude longer than that deduced from linear theories.

The Astrophysical Journal, 2008

We have investigated the formation of close-in extrasolar giant planets through a coupling effect... more We have investigated the formation of close-in extrasolar giant planets through a coupling effect of mutual scattering, Kozai mechanism, and tidal circularization, by orbital integrations. Close-in gas giants would have been originally formed at several AU's beyond the ice lines in protoplanetary disks and migrated close to their host stars. Although type II migration due to planet-disk interactions may be a major channel for the migration, we show that this scattering process would also give a non-negligible contribution. We have carried out orbital integrations of three planets with Jupiter-mass, directly including the effect of tidal circularization. We have found that in about 30% runs close-in planets are formed, which is much higher than suggested by previous studies. Three-planet orbit crossing usually results in one or two planets ejection. The tidal circularization often occurs during the three-planet orbit crossing, but previous studies have monitored only the final stage after the ejection, significantly underestimating the formation probability. We have found that Kozai mechanism by outer planets is responsible for the formation of close-in planets. During the three-planet orbital crossing, the Kozai excitation is repeated and the eccentricity is often increased secularly to values close enough to unity for tidal circularization to transform the inner planet to a close-in planet. Since a moderate eccentricity can remain for the close-in planet, this mechanism may account for the observed close-in planets with moderate eccentricities and without nearby secondary planets. Since these planets also remain a broad range of orbital inclinations (even retrograde ones), the contribution of this process would be clarified by more observations of Rossiter-McLaughlin effects for transiting planets.

The Astrophysical Journal, 2010

The Jovian regular satellite system mainly consists of four Galilean satellites that have similar... more The Jovian regular satellite system mainly consists of four Galilean satellites that have similar masses and are trapped in mutual mean motion resonances except for the outer satellite, Callisto. On the other hand, the Saturnian regular satellite system has only one big icy body, Titan, and a population of much smaller icy moons. We have investigated the origin of these major differences between the Jovian and Saturnian satellite systems by semi-analytically simulating the growth and orbital migration of proto-satellites in an accreting proto-satellite disk. We set up two different disk evolution/structure models that correspond to Jovian and Saturnian systems, by building upon previously developed models of an actively-supplied proto-satellite disk, the formation of gas giants, and observations of young stars. Our simulations extend previous models by including the (1) different termination timescales of gas infall onto the protosatellite disk and (2) different evolution of a cavity in the disk, between the Jovian and Saturnian systems. We have performed Monte Carlo simulations and show that in the case of the Jovian systems, four to five similar-mass satellites are likely to remain trapped in mean motion resonances. This orbital configuration is formed by type I * a log uniform distribution

Icarus, 2012

Many extrasolar planetary systems containing multiple super-Earths have been discovered. N-body s... more Many extrasolar planetary systems containing multiple super-Earths have been discovered. N-body simulations taking into account standard type-I planetary migration suggest that protoplanets are captured into mean-motion resonant orbits near the inner disk edge at which the migration is halted. Previous N-body simulations suggested that orbital stability of the resonant systems depends on number of the captured planets. In the unstable case, through close scattering and merging between planets, non-resonant multiple systems are finally formed. In this paper, we investigate the critical number of the resonantly trapped planets beyond which orbital instability occurs after disk gas depletion. We find that when the total number of planets (N) is larger than the critical number (N crit), crossing time that is a timescale of initiation of the orbital instability is similar to non-resonant cases, while the orbital instability never occurs within the orbital calculation time (10 8 Kepler time) for N ≤ N crit. Thus, the transition of crossing time across the critical number is drastic. When all the planets are trapped in 7:6 resonance of adjacent pairs, N crit = 4. We examine the dependence of the critical number of 4:3, 6:5 and 8:7 resonance by changing the orbital separation in mutual Hill radii and planetary mass. The critical number increases with increasing the orbital separation in mutual Hill radii with fixed planetary mass and increases with increasing planetary mass with fixed the orbital separation in mutual Hill radii. We also calculate the case of a system which is not composed of