Does the gas flow through a porous dust aggregate help its growth in a protoplanetary disk?

Sekiya and Takeda [2003, Earth Planets Space 55, 263-269] showed that the hydrodynamic gas flow around a dust aggregate larger than the mean free path of gas molecules prevents the growth of the body. In contrast to Wurm et al. [2004, Astrophys. J. 606, 983-987], we argue that this conclusion is not altered even if we take account of the effect of the flow through a porous dust aggregate.     

The gravitational instability in the dust layer of a protoplanetary disk:: axisymmetric solutions for nonuniform dust density distributions in the direction vertical to the midplane

The gravitational instability in the dust layer of a protoplanetary disk with nonuniform dust density distributions in the direction vertical to the midplane is investigated. The linear analysis of the gravitational instability is performed. The following assumptions are used: (1) One fluid model is adopted, that is, difference of velocities between dust and gas are neglected. (2) The gas is incompressible. (3) Models are axisymmetric with respect to the rotation axis of the disk. Numerical results show that the critical density at the midplane is higher than the one for the uniform dust density distribution by Sekiya (1983, Prog. Theor. Phys. 69, 1116-1130). For the Gaussian dust density distribution, the critical density is 1.3 times higher, although we do not consider this dust density distribution to be realistic because of the shear instability in the dust layer. For the dust density distribution with a constant Richardson number, which is considered to be realized due to the shear instability, the critical density is 2.85 times higher and is independent of the value of the Richardson number. Further, if a constant Richardson number could decrease to the order of 0.001, the gravitational instability would be realized even for the dust to gas surface density ratio with the solar abundance. Our results give a new restriction on planetesimal formation by the gravitational instability.     

Accretion rates of planetesimals by protoplanets embedded in nebular gas

When protoplanets growing by accretion of planetesimals have atmospheres, small planetesimals approaching the protoplanets lose their energy by gas drag from the atmospheres, which leads them to be captured within the Hill sphere of the protoplanets. As a result, growth rates of the protoplanets are enhanced. In order to study the effect of an atmosphere on planetary growth rates, we performed numerical integration of orbits of planetesimals for a wide range of orbital elements and obtained the effective accretion rates of planetesimals onto planets that have atmospheres. Numerical results are obtained as a function of planetesimals’ eccentricity, inclination, planet’s radius, and non-dimensional gas-drag parameters which can be expressed by several physical quantities such as the radius of planetesimals and the mass of the protoplanet. Assuming that the radial distribution of the gas density near the surface can be approximated by a power-law, we performed analytic calculation for the loss of planetesimals’ kinetic energy due to gas drag, and confirmed agreement with numerical results. We confirmed that the above approximation of the power-law density distribution is reasonable for accretion rate of protoplanets with 1-10 Earth masses, unless the size of planetesimals is too small. We also calculated the accretion rates of planetesimals averaged over a Rayleigh distribution of eccentricities and inclinations, and derived a semi-analytical formula of accretion rates, which reproduces the numerical results very well. Using the obtained expression of the accretion rate, we examined the growth of protoplanets in nebular gas. We found that the effect of atmospheric gas drag can enhance the growth rate significantly, depending on the size of planetesimals.     

Metallicity of the massive protoplanets around HR 8799 If formed by gravitational instability

The final composition of giant planets formed as a result of gravitational instability in the disk gas depends on their ability to capture solid material (planetesimals) during their ‘pre-collapse’ stage, when they are extended and cold, and contracting quasi-statically. The duration of the pre-collapse stage is inversely proportional roughly to the square of the planetary mass, so massive protoplanets have shorter pre-collapse timescales and therefore limited opportunity for planetesimal capture. The available accretion time for protoplanets with masses of 3, 5, 7, and 10 Jupiter masses is found to be and 5.67×10³ years, respectively. The total mass that can be captured by the protoplanets depends on the planetary mass, planetesimal size, the radial distance of the protoplanet from the parent star, and the local solid surface density. We consider three radial distances, 24, 38, and 68 AU, similar to the radial distances of the planets in the system HR 8799, and estimate the mass of heavy elements that can be accreted. We find that for the planetary masses usually adopted for the HR 8799 system, the amount of heavy elements accreted by the planets is small, leaving them with nearly stellar compositions.     

The dynamics of Jupiter and Saturn in the gaseous protoplanetary disk

We study the possibility that the mutual interactions between Jupiter and Saturn prevented Type II migration from driving these planets much closer to the Sun. Our work extends previous results by Masset and Snellgrove [Masset, F., Snellgrove, M., 2001. Mon. Not. R. Astron. Soc. 320, L55-L59], by exploring a wider set of initial conditions and disk parameters, and by using a new hydrodynamical code that properly describes for the global viscous evolution of the disk. Initially both planets migrate towards the Sun, and Saturn's migration tends to be faster. As a consequence, they eventually end up locked in a mean motion resonance. If this happens in the 2:3 resonance, the resonant motion is particularly stable, and the gaps opened by the planets in the disk may overlap. This causes a drastic change in the torque balance for the two planets, which substantially slows down the planets' inward migration. If the gap overlap is substantial, planet migration may even be stopped or reversed. As the widths of the gaps depend on disk viscosity and scale height, this mechanism is particularly efficient in low viscosity, cool disks. The initial locking of the planets in the 2:3 resonance is a likely outcome if Saturn formed at the edge of Jupiter's gap, but also if Saturn initially migrated rapidly from further away. We also explore the possibility of trapping in other resonances, and the subsequent evolutions. We discuss the compatibility of our results with the initial conditions adopted in Tsiganis et al. [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461] and Gomes et al. [Gomes, R., Levison, H.F., Tsiganis, K., Morbidelli, A., 2005. Nature 435, 466-469] to explain the current orbital architecture of the giant planets and the origin of the Late Heavy Bombardment of the Moon.     

Deciphering the origin of the regular satellites of gaseous giants - Iapetus: The Rosetta ice-moon

Ever since their discovery the regular satellites of Jupiter and Saturn have held out the promise of providing an independent set of observations with which to test theories of planet formation. Yet elucidating their origins has proven elusive. Here we show that Iapetus can serve to discriminate between satellite formation models. Its accretion history can be understood in terms of a two-component gaseous subnebula, with a relatively dense inner region, and an extended tail out to the location of the irregular satellites, as in the SEMM model of Mosqueira and Estrada (2003a,b) (Mosqueira, I., Estrada, P.R. [2003a]. Icarus 163, 198-231; Mosqueira, I., Estrada, P.R. [2003b]. Icarus 163, 232-255). Following giant planet formation, planetesimals in the feeding zone of Jupiter and Saturn become dynamically excited, and undergo a collisional cascade. Ablation and capture of planetesimal fragments crossing the gaseous circumplanetary disks delivers enough collisional rubble to account for the mass budgets of the regular satellites of Jupiter and Saturn. This process can result in rock/ice fractionation as long as the make up of the population of disk crossers is non-homogeneous, thus offering a natural explanation for the marked compositional differences between outer solar nebula objects and those that accreted in the subnebulae of the giant planets. For a given size, icy objects are easier to capture and to ablate, likely resulting in an overall enrichment of ice in the subnebula. Furthermore, capture and ablation of rocky fragments become inefficient far from the planet for two reasons: the gas surface density of the subnebula is taken to drop outside the centrifugal radius, and the velocity of interlopers decreases with distance from the planet. Thus, rocky objects crossing the outer disks of Jupiter and Saturn never reach a temperature high enough to ablate either due to melting or vaporization, and capture is also greatly diminished there. In contrast, icy objects crossing the outer disks of each planet ablate due to the melting and vaporization of water-ice. Consequently, our model leads to an enhancement of the ice content of Iapetus, and to a lesser degree those of Titan, Callisto and Ganymede, and accounts for the (non-stochastic) compositions of these large, low-porosity outer regular satellites of Jupiter and Saturn. For this to work, the primordial population of planetesimals in the Jupiter-Saturn region must be partially differentiated, so that the ensuing collisional cascade produces an icy population of ?1 m size fragments to be ablated during subnebula crossing. We argue this is likely because the first generation of solar nebula ∼10 km planetesimals in the Jupiter-Saturn region incorporated significant quantities of ²⁶Al. This is the first study successfully to provide a direct connection between nebula planetesimals and subnebulae mixtures with quantifiable and observable consequences for the bulk properties of the regular satellites of Jupiter and Saturn, and the only explanation presently available for Iapetus’ low density and ice-rich composition.     

Simulations of planet migration driven by planetesimal scattering

Evidence has mounted for some time that planet migration is an important part of the formation of planetary systems, both in the Solar System [Malhotra, R., 1993. Nature 365, 819-821] and in extrasolar systems [Mayor, M., Queloz, D., 1995. Nature 378, 355-359; Lin, D.N.C., Bodenheimer, P., Richardson, D.C., 1996. Nature 380, 606-607]. One mechanism that produces migration (the change in a planet's semi-major axis a over time) is the scattering of comet- and asteroid-size bodies called planetesimals [Fernandez, J.A., Ip, W.-H., 1984. Icarus 58, 109-120]. Significant angular momentum exchange can occur between the planets and the planetesimals during local scattering, enough to cause a rapid, self-sustained migration of the planet [Ida, S., Bryden, G., Lin, D.N.C., Tanaka, H., 2000. Astrophys. J. 534, 428-445]. This migration has been studied for the particular case of the four outer planets of the Solar System (as in Gomes et al. [Gomes, R.S., Morbidelli, A., Levison, H.F., 2004. Icarus 170, 492-507]), but is not well understood in general. We have used the Miranda [McNeil, D., Duncan, M., Levison, H.F., 2005. Astron. J. 130, 2884-2899] computer simulation code to perform a broad parameter-space survey of the physical variables that determine the migration of a single planet in a planetesimal disk. Migration is found to be predominantly inwards, and the migration rate is found to be independent of planet mass for low-mass planets in relatively high-mass disks. Indeed, a simple scaling relation from Ida et al. [Ida, S., Bryden, G., Lin, D.N.C., Tanaka, H., 2000. Astrophys. J. 534, 428-445] matches well with the dependencies of the migration rate:

(1)     

Particle stirring in turbulent gas disks: Including orbital oscillations

We describe the diffusion and random velocities of solid particles due to stochastic forcing by turbulent gas. We include the orbital dynamics of Keplerian disks, both in-plane epicycles and vertical oscillations. We obtain a new result for the diffusion of solids. The Schmidt number (ratio of gas to particle diffusivity) is Sc≈1+(Ωt_stop)², in terms of the particle stopping time t_stop and the orbital frequency Ω. The standard result, Sc=1+t_stop/t_eddy, in terms of the eddy turnover time, t_eddy, is shown to be incorrect. The main difference is that Sc rises quadratically, not linearly, with stopping time. Consequently, particles larger than 10 cm in protoplanetary disks will suffer less radial diffusion and will settle closer to the midplane. Such a layer of boulders would be more prone to gravitational collapse. Our predictions of RMS speeds, vertical scale height and diffusion coefficients will help interpret numerical simulations. We confirm previous results for the vertical stirring of particles (scale heights and random velocities), and add a correction for arbitrary ratios of eddy to orbital times. The particle layer becomes thinner for t_eddy>1/Ω with the strength of turbulent diffusion held fixed. We use two analytic techniques—the Hinze–Tchen formalism and the Fokker–Planck equation with velocity diffusion—with identical results when the regimes of validity overlap. We include simple physical arguments for the scaling of our results.     

Oligarchic growth with migration and fragmentation

In the core-accretion model, giant-planet cores form by oligarchic growth from a population of planetesimals prior to the dispersal of the disk gas. Once a core reaches a critical mass of roughly 10 Earth masses, it begins to accrete a gaseous envelope, forming a giant planet. Collisions between planetesimals cause fragmentation. Planetesimal fragments are more easily captured by cores, speeding up growth, but fragments are also lost by radial drift, reducing the total solid mass in the disk. Interaction with the gas causes cores to undergo inward type-I migration. Migration allows a core to accrete planetesimals from a larger region, but migrating cores may be lost if they reach the star. Thus, migration and fragmentation have both a positive and a negative impact on core formation. Here we describe results of new simulations of oligarchic growth that include fragmentation and/or migration. In the absence of migration, cores grow until they reach their isolation mass, which increases with distance from the star, or until the disk gas disperses. Fragmentation increases the maximum core mass by increasing growth rates in the outer disk, allowing objects to reach their isolation mass during the disk lifetime. When migration is present, cores migrate inwards rapidly when they approach 1 Earth mass. Most migrating cores are lost. Migrating cores gain little extra mass since they are passing through regions that have been depleted by earlier generations of cores. For a disk viscosity parameter alpha=1e−3 and planetesimal radius = 10 km, the maximum core mass is roughly 4 and 0.5 Earth masses with/without fragmentation, respectively, with little dependence on the disk mass. Formation and survival of 10-Earth-mass cores, in the presence of migration, requires large alpha (1e−2) and a massive disk (0.1 solar masses). When alpha is large, type-I migration rates decrease rapidly with time, allowing large, late-forming cores to survive. The addition of a stochastic (random-walk) migration component makes little difference to the outcome, provided that stochastic migration affects only cores larger than 0.01 Earth masses. Stochastic migration becomes increasingly important if it also affects lower-mass objects.     

Planetesimal capture in the disk instability model

We follow the contraction and evolution of a typical Jupiter-mass clump created by the disk instability mechanism, and compute the rate of planetesimal capture during this evolution. We show that such a clump has a slow contraction phase lasting ∼3×¹⁰⁵ yr. By following the trajectories of planetesimals as they pass through the envelope of the protoplanet, we compute the cross-section for planetesimal capture at all stages of the protoplanet's evolution. We show that the protoplanet can capture a large fraction of the solid material in its feeding zone, which will lead to an enrichment of the protoplanet in heavy elements. The exact amount of this enrichment depends upon, but is not very sensitive to the size and random speed of the planetesimals.     

Orbital evolution and accretion of protoplanets tidally interacting with a gas disk: I. Effects of interaction with planetesimals and other protoplanets

We have performed N-body simulations on the stage of protoplanet formation from planetesimals, taking into account so-called “type-I migration,” and damping of orbital eccentricities and inclinations, as a result of tidal interaction with a gas disk without gap formation. One of the most serious problems in formation of terrestrial planets and jovian planet cores is that the migration time scale predicted by the linear theory is shorter than the disk lifetime (10⁶-10⁷ years). In this paper, we investigate retardation of type-I migration of a protoplanet due to a torque from a planetesimal disk in which a gap is opened up by the protoplanet, and torques from other protoplanets which are formed in inner and outer regions. In the first series of runs, we carried out N-body simulations of the planetesimal disk, which ranges from 0.9 to 1.1 AU, with a protoplanet seed in order to clarify how much retardation can be induced by the planetesimal disk and how long such retardation can last. We simulated six cases with different migration speeds. We found that in all of our simulations, a clear gap is not maintained for more than 10⁵ years in the planetesimal disk. For very fast migration, a gap cannot be created in the planetesimal disk. For migration slower than some critical speed, a gap does form. However, because of the growth of the surrounding planetesimals, gravitational perturbation of the planetesimals eventually becomes so strong that the planetesimals diffuse into the vicinity of the protoplanets, resulting in destruction of the gap. After the gap is destroyed, close encounters with the planetesimals rather accelerate the protoplanet migration. In this way, the migration cannot be retarded by the torque from the planetesimal disk, regardless of the migration speed. In the second series of runs, we simulated accretion of planetesimals in wide range of semimajor axis, 0.5 to 2-5 AU, starting with equal mass planetesimals without a protoplanet seed. Since formation of comparable-mass multiple protoplanets (“oligarchic growth”) is expected, the interactions with other protoplanets have a potential to alter the migration speed. However, inner protoplanets migrate before outer ones are formed, so that the migration and the accretion process of a runaway protoplanet are not affected by the other protoplanets placed inner and outer regions of its orbit. From the results of these two series of simulations, we conclude that the existence of planetesimals and multiple protoplanets do not affect type-I migration and therefore the migration shall proceed as the linear theory has suggested.     

Orbital evolution and accretion of protoplanets tidally interacting with a gas disk: II. Solid surface density evolution with type-I migration

This paper investigates the surface density evolution of a planetesimal disk due to the effect of type-I migration by carrying out N-body simulation and through analytical method, focusing on terrestrial planet formation. The coagulation and the growth of the planetesimals take place in the abundant gas disk except for a final stage. A protoplanet excites density waves in the gas disk, which causes the torque on the protoplanet. The torque imbalance makes the protoplanet suffer radial migration, which is known as type-I migration. Type-I migration time scale derived by the linear theory may be too short for the terrestrial planets to survive, which is one of the major problems in the planet formation scenario. Although the linear theory assumes a protoplanet being in a gas disk alone, Kominami et al. [Kominami, J., Tanaka, H., Ida, S., 2005. Icarus 167, 231-243] showed that the effect of the interaction with the planetesimal disk and the neighboring protoplanets on type-I migration is negligible. The migration becomes pronounced before the planet's mass reaches the isolation mass, and decreases the solid component in the disk. Runaway protoplanets form again in the planetesimal disk with decreased surface density. In this paper, we present the analytical formulas that describe the evolution of the solid surface density of the disk as a function of gas-to-dust ratio, gas depletion time scale and semimajor axis, which agree well with our results of N-body simulations. In general, significant depletion of solid material is likely to take place in inner regions of disks. This might be responsible for the fact that there is no planet inside Mercury's orbit in our Solar System. Our most important result is that the final surface density of solid components (Σd) and mass of surviving planets depend on gas surface density (Σg) and its depletion time scale (τdep) but not on initial Σd; they decrease with increase in Σg and τdep. For a fixed gas-to-dust ratio and τdep, larger initial Σd results in smaller final Σd and smaller surviving planets, because of larger Σg. To retain a specific amount of Σd, the efficient disk condition is not an initially large Σd but the initial Σd as small as the specified final one and a smaller gas-to-dust ratio. To retain Σd comparable to that of the minimum mass solar nebula (MMSN), a disk must have the same Σd and a gas-to-dust ratio that is smaller than that of MMSN by a factor of 1.3×(τdep/1 Myr) at ∼1 AU. (Equivalently, type-I migration speed is slower than that predicted by the linear theory by the same factor.) The surviving planets are Mars-sized ones in this case; in order to form Earth-sized planets, their eccentricities must be pumped up to start orbit crossing and coagulation among them. At ∼5 AU, Σd of MMSN is retained under the same condition, but to form a core massive enough to start runaway gas accretion, a gas-to-dust ratio must be smaller than that of MMSN by a factor of 3×τdep/1 Myr.     

Forming Jupiter, Saturn, Uranus and Neptune in few million years by core accretion

Giant planet formation process is still not completely understood. The current most accepted paradigm, the core instability model, explains several observed properties of the Solar System’s giant planets but, to date, has faced difficulties to account for a formation time shorter than the observational estimates of protoplanetary disks’ lifetimes, especially for the cases of Uranus and Neptune. In the context of this model, and considering a recently proposed primordial Solar System orbital structure, we performed numerical calculations of giant planet formation. Our results show that if accreted planetesimals follow a size distribution in which most of the mass lies in 30-100 m sized bodies, Jupiter, Saturn, Uranus and Neptune may have formed according to the nucleated instability scenario. The formation of each planet occurs within the time constraints and they end up with core masses in good agreement with present estimations.     

Formation of gas giant planets: core accretion models with fragmentation and planetary envelope

We have calculated formation of gas giant planets based on the standard core accretion model including effects of fragmentation and planetary envelope. The accretion process is found to proceed as follows. As a result of runaway growth of planetesimals with initial radii of ∼10 km, planetary embryos with a mass of ∼10²⁷ g (∼ Mars mass) are found to form in ∼10⁵ years at Jupiter's position (5.2 AU), assuming a large enough value of the surface density of solid material (25 g/cm²) in the accretion disk at that distance. Strong gravitational perturbations between the runaway planetary embryos and the remaining planetesimals cause the random velocities of the planetesimals to become large enough for collisions between small planetesimals to lead to their catastrophic disruption. This produces a large number of fragments. At the same time, the planetary embryos have envelopes, that reduce energies of fragments by gas drag and capture them. The large radius of the envelope increases the collision rate between them, resulting in rapid growth of the planetary embryos. By the combined effects of fragmentation and planetary envelope, the largest planetary embryo with 21M_⊕ forms at 5.2 AU in 3.8×10⁶ years. The planetary embryo is massive enough to start a rapid gas accretion and forms a gas giant planet.     

Accretion of terrestrial planets from oligarchs in a turbulent disk

We have investigated the final accretion stage of terrestrial planets from Mars-mass protoplanets that formed through oligarchic growth in a disk comparable to the minimum mass solar nebula (MMSN), through N-body simulation including random torques exerted by disk turbulence due to Magneto-Rotational Instability. For the torques, we used the semi-analytical formula developed by Laughlin et al. [Laughlin, G., Steinacker, A., Adams, F.C., 2004. Astrophys. J. 608, 489-496]. The damping of orbital eccentricities (in all runs) and type-I migration (in some runs) due to the tidal interactions with disk gas is also included. Without any effect of disk gas, Earth-mass planets are formed in terrestrial planet regions in a disk comparable to MMSN but with too large orbital eccentricities to be consistent with the present eccentricities of Earth and Venus in our Solar System. With the eccentricity damping caused by the tidal interaction with a remnant gas disk, Earth-mass planets with eccentricities consistent with those of Earth and Venus are formed in a limited range of disk gas surface density (∼10⁻⁴ times MMSN). However, in this case, on average, too many (?6) planets remain in terrestrial planet regions, because the damping leads to isolation between the planets. We have carried out a series of N-body simulations including the random torques with different disk surface density and strength of turbulence. We found that the orbital eccentricities pumped up by the turbulent torques and associated random walks in semimajor axes tend to delay isolation of planets, resulting in more coagulation of planets. The eccentricities are still damped after planets become isolated. As a result, the number of final planets decreases with increase in strength of the turbulence, while Earth-mass planets with small eccentricities are still formed. In the case of relatively strong turbulence, the number of final planets are 4-5 at 0.5-2 AU, which is more consistent with Solar System, for relatively wide range of disk gas surface density (∼¹⁰⁻⁴-¹⁰⁻² times MMSN).     

Planetary growth with collisional fragmentation and gas drag

As planetary embryos grow, gravitational stirring of planetesimals by embryos strongly enhances random velocities of planetesimals and makes collisions between planetesimals destructive. The resulting fragments are ground down by successive collisions. Eventually the smallest fragments are removed by the inward drift due to gas drag. Therefore, the collisional disruption depletes the planetesimal disk and inhibits embryo growth. We provide analytical formulae for the final masses of planetary embryos, taking into account planetesimal depletion due to collisional disruption. Furthermore, we perform the statistical simulations for embryo growth (which excellently reproduce results of direct N-body simulations if disruption is neglected). These analytical formulae are consistent with the outcome of our statistical simulations. Our results indicate that the final embryo mass at several AU in the minimum-mass solar nebula can reach about ∼0.1 Earth mass within 10⁷ years. This brings another difficulty in formation of gas giant planets, which requires cores with ∼10 Earth masses for gas accretion. However, if the nebular disk is 10 times more massive than the minimum-mass solar nebula and the initial planetesimal size is larger than 100 km, as suggested by some models of planetesimal formation, the final embryo mass reaches about 10 Earth masses at 3-4 AU. The enhancement of embryos’ collisional cross sections by their atmosphere could further increase their final mass to form gas giant planets at 5-10 AU in the Solar System.     

Experimental study on the impact fragmentation of core-mantle bodies: Implications for collisional disruption of rocky planetesimals with sintered core covered with porous mantle

Impact experiments of inhomogeneous targets such as layered bodies consisting of a dense core and porous mantle were conducted to clarify the effect of the layered structure on impact strength. The layered structure of small bodies could be the result of the thermal evolution of planetesimals in the solar nebula. So, the impact disruption of thermally evolved bodies with core-mantle structure is important for the origin of small bodies such as asteroids. We investigated the impact strength of rocky-layered bodies with porous mantle-sintered cores, which could be formed at an initial stage of thermal evolution. Spherical targets composed of soda-lime glass or quartz core and porous gypsum mantle were prepared as an analog of small bodies with a core-mantle structure, and the internal structure was changed. A nylon projectile was impacted at the impact velocity from 1 to 5 km/s. The impact strength of the core-mantle targets decreases with the increase of the core/target mass ratio (RCM) in the specific energy range from 1×¹⁰³ to 4×¹⁰⁴ J/kg. We observed two distinct destruction modes characterized by the damage to the core: one shows a damaged core and fractured mantle, and the other shows an intact core and broken mantle. The former mode was usually observed with increasing RCM, and the boundary condition of the core destruction () was experimentally found to be , where is the specific energy required to disrupt a glass core. From this empirical equation, it might be possible to discuss the destruction conditions of a thermally evolved body with a porous mantle-sintered core structure. We speculate that the impact strength of the body could be significantly reduced with the progress of internal evolution at the initial stage of thermal evolution.     

Transformation of Trojans into quasi-satellites during planetary migration and their subsequent close-encounters with the host planet

We use numerical integrations to investigate the dynamical evolution of resonant Trojan and quasi-satellite companions during the late stages of migration of the giant planets Jupiter, Saturn, Uranus, and Neptune. Our migration simulations begin with Jupiter and Saturn on orbits already well separated from their mutual 2:1 mean-motion resonance. Neptune and Uranus are decoupled from each other and have orbital eccentricities damped to near their current values. From this point we adopt a planet migration model in which the migration speed decreases exponentially with a characteristic timescale τ (the e-folding time). We perform a series of numerical simulations, each involving the migrating giant planets plus test particle Trojans and quasi-satellites. We find that the libration frequencies of Trojans are similar to those of quasi-satellites. This similarity enables a dynamical exchange of objects back and forth between the Trojan and quasi-satellite resonances during planetary migration. This exchange is facilitated by secondary resonances that arise whenever there is more than one migrating planet. For example, secondary resonances may occur when the circulation frequencies, f, of critical arguments for the Uranus-Neptune 2:1 mean-motion near-resonance are commensurate with harmonics of the libration frequency of the critical argument for the Trojan and quasi-satellite 1:1 mean-motion resonance . Furthermore, under the influence of these secondary resonances quasi-satellites can have their libration amplitudes enlarged until they undergo a close-encounter with their host planet and escape from the resonance. High-resolution simulations of this escape process reveal that ≈80% of jovian quasi-satellites experience one or more close-encounters within Jupiter’s Hill radius (R_H) as they are forced out of the quasi-satellite resonance. As many as ≈20% come within R_H/4 and ≈2.5% come within R_H/10. Close-encounters of escaping quasi-satellites occur near or even below the 2-body escape velocity from the host planet. Finally, the exchange and escape of Trojans and quasi-satellites continues to as late as 6-9τ in some simulations. By this time the dynamical evolution of the planets is strongly dominated by distant gravitational perturbations between the planets rather than the migration force. This suggests that exchange and escape of Trojans and quasi-satellites may be a contemporary process associated with the present-day near-resonant configuration of some of the giant planets in our Solar System.     

Evolution of the obliquities of the giant planets in encounters during migration

Tsiganis et al. [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461] have proposed that the current orbital architecture of the outer Solar System could have been established if it was initially compact and Jupiter and Saturn crossed the 2:1 orbital resonance by divergent migration. The crossing led to close encounters among the giant planets, but the orbital eccentricities and inclinations were damped to their current values by interactions with planetesimals. Brunini [Brunini, A., 2006. Nature 440, 1163-1165] has presented widely publicized numerical results showing that the close encounters led to the current obliquities of the giant planets. We present a simple analytic argument which shows that the change in the spin direction of a planet relative to an inertial frame during an encounter between the planets is very small and that the change in the obliquity (which is measured from the orbit normal) is due to the change in the orbital inclination. Since the inclinations are damped by planetesimal interactions on timescales much shorter than the timescales on which the spins precess due to the torques from the Sun, especially for Uranus and Neptune, the obliquities should return to small values if they are small before the encounters. We have performed simulations using the symplectic integrator SyMBA, modified to include spin evolution due to the torques from the Sun and mutual planetary interactions. Our numerical results are consistent with the analytic argument for no significant remnant obliquities.     

Accretion among preplanetary bodies: The many faces of runaway growth

When preplanetary bodies reach proportions of ∼1 km or larger in size, their accretion rate is enhanced due to gravitational focusing (GF). We have developed a new numerical model to calculate the collisional evolution of the gravitationally-enhanced growth stage. The numerical model is novel as it attempts to preserve the individual particle nature of the bodies (like N-body codes); yet it is statistical in nature since it must incorporate the very large number of planetesimals. We validate our approach against existing N-body and statistical codes. Using the numerical model, we explore the characteristics of the runaway growth and the oligarchic growth accretion phases starting from an initial population of single planetesimal radius R₀. In models where the initial random velocity dispersion (as derived from their eccentricity) starts out below the escape speed of the planetesimal bodies, the system experiences runaway growth. We associate the initial runaway growth phase with increasing GF-factors for the largest body. We find that during the runaway growth phase the size distribution remains continuous but evolves into a power-law at the high-mass end, consistent with previous studies. Furthermore, we find that the largest body accretes from all mass bins; a simple two-component approximation is inapplicable during this stage. However, with growth the runaway body stirs up the random motions of the planetesimal population from which it is accreting. Ultimately, this feedback stops the fast growth and the system passes into oligarchy, where competitor bodies from neighboring zones catch up in terms of mass. We identify the peak of GF with the transition between the runaway growth and oligarchy accretion stages. Compared to previous estimates, we find that the system leaves the runaway growth phase at a somewhat larger radius, especially at the outer disk. Furthermore, we assess the relevance of small, single-size fragments on the growth process. In classical models, where the initial velocity dispersion of bodies is small, these do not play a critical role during the runaway growth; however, in models that are characterized by large initial relative velocities due to external stirring of their random motions, a situation can emerge where fragments dominate the accretion, which could lead to a very fast growth.