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).     

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.     

Formation of terrestrial planets in a dissipating gas disk with Jupiter and Saturn

We have performed N-body simulations on final accretion stage of terrestrial planets, including the eccentricity and inclination damping effect due to tidal interaction with a gas disk. We investigated the dependence on a depletion time scale of the disk, and the effect of secular perturbations by Jupiter and Saturn. In the final stage, terrestrial planets are formed through coagulation of protoplanets of about the size of Mars. They would collide and grow in a decaying gas disk. Kominami and Ida [Icarus 157 (2002) 43-56] showed that it is plausible that Earth-sized, low-eccentricity planets are formed in a mostly depleted gas disk. In this paper, we investigate the formation of planets in a decaying gas disk with various depletion time scales, assuming disk surface density of gas component decays exponentially with time scale of τ_gas. Fifteen protoplanets with are initially distributed in the terrestrial planet regions. We found that Earth-sized planets with low eccentricities are formed, independent of initial gas surface density, when the condition (τ_cross+τ_growth)/2?τ_gas?τ_cross is satisfied, where τ_cross is the time scale for initial protoplanets to start orbit crossing in a gas-free case and τ_growth is the time scale for Earth-sized planets to accrete during the orbit crossing stage. In the cases satisfying the above condition, the final masses and eccentricities of the largest planets are consistent with those of Earth and Venus. However, four or five protoplanets with the initial mass remain. In the final stage of terrestrial planetary formation, it is likely that Jupiter and Saturn have already been formed. When Jupiter and Saturn are included, their secular perturbations pump up eccentricities of protoplanets and tend to reduce the number of final planets in the terrestrial planet regions. However, we found that the reduction is not significant. The perturbations also shorten τ_cross. If the eccentricities of Jupiter and Saturn are comparable to or larger than present values (∼0.05), τ_cross become too short to satisfy the above condition. As a result, eccentricities of the planets cannot be damped to the observed value of Earth and Venus. Hence, for the formation of terrestrial planets, it is preferable that the secular perturbations from Jupiter and Saturn do not have significant effect upon the evolution. Such situation may be reproduced by Jupiter and Saturn not being fully grown, or their eccentricities being smaller than the present values during the terrestrial planets' formation. However, in such cases, we need some other mechanism to eliminate the problem that numerous Mars-sized planets remain uncollided.     

Making other earths: dynamical simulations of terrestrial planet formation and water delivery

We present results from 44 simulations of late stage planetary accretion, focusing on the delivery of volatiles (primarily water) to the terrestrial planets. Our simulations include both planetary “embryos” (defined as Moon to Mars sized protoplanets) and planetesimals, assuming that the embryos formed via oligarchic growth. We investigate volatile delivery as a function of Jupiter's mass, position and eccentricity, the position of the snow line, and the density (in solids) of the solar nebula. In all simulations, we form 1-4 terrestrial planets inside 2 AU, which vary in mass and volatile content. In 44 simulations we have formed 43 planets between 0.8 and 1.5 AU, including 11 “habitable” planets between 0.9 and 1.1 AU. These planets range from dry worlds to “water worlds” with 100+oceans of water (1 ocean=1.5×10²⁴ g), and vary in mass between 0.23M_⊕ and 3.85M_⊕. There is a good deal of stochastic noise in these simulations, but the most important parameter is the planetesimal mass we choose, which reflects the surface density in solids past the snow line. A high density in this region results in the formation of a smaller number of terrestrial planets with larger masses and higher water content, as compared with planets which form in systems with lower densities. We find that an eccentric Jupiter produces drier terrestrial planets with higher eccentricities than a circular one. In cases with Jupiter at 7 AU, we form what we call “super embryos,” 1-2M_⊕ protoplanets which can serve as the accretion seeds for 2+M_⊕ planets with large water contents.     

An efficient, low-velocity, resonant mechanism for capture of satellites by a protoplanet

Numerical simulations of the gravitational scattering of planetesimals by a protoplanet reveal that a significant fraction of scattered planetesimals can become trapped as so-called quasi-satellites in heliocentric 1:1 co-orbital resonance with the protoplanet. While trapped, these resonant planetesimals can have deep low-velocity encounters with the protoplanet that result in temporary or permanent capture onto highly eccentric prograde or retrograde circumplanetary orbits. The simulations include solar nebula gas drag and use planetesimals with diameters ranging from ∼1 to ∼1000 km. Initial protoplanet eccentricities range from ep=0 to 0.15 and protoplanet masses range from 300 Earth-masses (M_⊕) down to 0.1M_⊕. This mass range effectively covers the final masses of all planets currently thought to be in possession of captured satellites—Jupiter, Saturn, Neptune, Uranus, and Mars. For protoplanets on moderately eccentric orbits (ep?0.1) most simulations show from 5-20% of all scattered planetesimals becoming temporarily trapped in the quasi-satellite co-orbital resonance. Typically, 20-30% of the temporarily trapped quasi-satellites of all sizes came within half the Hill radius of the protoplanet while trapped in the resonance. The efficiency of the resonance trapping combined with the subsequent low-velocity circumplanetary capture suggests that this trapped-to-captured transition may be important not only for the origin of captured satellites but also for continued growth of protoplanets.     

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.     

Core formation in giant gaseous protoplanets

Sedimentation rates of silicate grains in gas giant protoplanets formed by disk instability are calculated for protoplanetary masses between 1 MSaturn to 10 MJupiter. Giant protoplanets with masses of 5 MJupiter or larger are found to be too hot for grain sedimentation to form a silicate core. Smaller protoplanets are cold enough to allow grain settling and core formation. Grain sedimentation and core formation occur in the low mass protoplanets because of their slow contraction rate and low internal temperature. It is predicted that massive giant planets will not have cores, while smaller planets will have small rocky cores whose masses depend on the planetary mass, the amount of solids within the body, and the disk environment. The protoplanets are found to be too hot to allow the existence of icy grains, and therefore the cores are predicted not to contain any ices. It is suggested that the atmospheres of low mass giant planets are depleted in refractory elements compared with the atmospheres of more massive planets. These predictions provide a test of the disk instability model of gas giant planet formation. The core masses of Jupiter and Saturn were found to be ∼0.25 M_⊕ and ∼0.5 M_⊕, respectively. The core masses of Jupiter and Saturn can be substantially larger if planetesimal accretion is included. The final core mass will depend on planetesimal size, the time at which planetesimals are formed, and the size distribution of the material added to the protoplanet. Jupiter's core mass can vary from 2 to 12 M_⊕. Saturn's core mass is found to be ∼8 M_⊕.     

Eccentricity evolution of giant planet orbits due to circumstellar disk torques

The extrasolar planets discovered to date possess unexpected orbital elements. Most orbit their host stars with larger eccentricities and smaller semi-major axes than similarly sized planets in our own Solar System do. It is generally agreed that the interaction between giant planets and circumstellar disks (Type II migration) drives these planets inward to small radii, but the effect of these same disks on orbital eccentricity, ?, is controversial. Several recent analytic calculations suggest that disk-planet interactions can excite eccentricity, while numerical studies generally produce eccentricity damping. This paper addresses this controversy using a quasi-analytic approach, drawing on several preceding analytic studies. This work refines the current treatment of eccentricity evolution by removing several approximations from the calculation of disk torques. We encounter neither uniform damping nor uniform excitation of orbital eccentricity, but rather a function d?/dt that varies in both sign and magnitude depending on eccentricity and other Solar System properties. Most significantly, we find that for every combination of disk and planet properties investigated herein, corotation torques produce negative values of d?/dt for some range in ? within the interval [0.1, 0.5]. If corotation torques are saturated, this region of eccentricity damping disappears, and excitation occurs on a short timescale of less than 0.08 Myr. Thus, our study does not produce eccentricity excitation on a timescale of a few Myr—we obtain either eccentricity excitation on a short time scale, or eccentricity damping on a longer time scale. Finally, we discuss the implications of this result for producing the observed range in extrasolar planet eccentricity.     

Eccentric Extrasolar Planets: The Jumping Jupiter Model

Most extrasolar planets discovered to date are more massive than Jupiter, in surprisingly small orbits (semimajor axes less than 3 AU). Many of these have significant orbital eccentricities. Such orbits may be the product of dynamical interactions in multiplanet systems. We examine outcomes of such evolution in systems of three Jupiter-mass planets around a solar-mass star by integration of their orbits in three dimensions. Such systems are unstable for a broad range of initial conditions, with mutual perturbations leading to crossing orbits and close encounters. The time scale for instability to develop depends on the initial orbital spacing; some configurations become chaotic after delays exceeding 10⁸ y. The most common outcome of gravitational scattering by close encounters is hyperbolic ejection of one planet. Of the two survivors, one is moved closer to the star and the other is left in a distant orbit; for systems with equal-mass planets, there is no correlation between initial and final orbital positions. Both survivors may have significant eccentricities, and the mutual inclination of their orbits can be large. The inner survivor's semimajor axis is usually about half that of the innermost starting orbit. Gravitational scattering alone cannot produce the observed excess of “hot Jupiters” in close circular orbits. However, those scattered planets with large eccentricities and small periastron distances may become circularized if tidal dissipation is effective. Most stars with a massive planet in an eccentric orbit should have at least one additional planet of comparable mass in a more distant orbit.     

The origin of the Kuiper Belt high-inclination population

I simulate the orbital evolution of the four major planets and a massive primordial planetesimal disk composed of 10⁴ objects, which perturb the planets but not themselves. As Neptune migrates by energy and angular momentum exchange with the planetesimals, a large number of primordial Neptune-scattered objects are formed. These objects may experience secular, Kozai, and mean motion resonances that induce temporary decrease of their eccentricities. Because planets are migrating, some planetesimals can escape those resonances while in a low-eccentricity incursion, thus avoiding the return path to Neptune close encounter dynamics. In the end, this mechanism produces stable orbits with high inclination and moderate eccentricities. The population so formed together with the objects coming from the classical resonance sweeping process, originates a bimodal distribution for the Kuiper Belt orbits. The inclinations obtained by the simulations can attain values above 30° and their distribution resembles a debiased distribution for the high-inclination population coming from the real classical Kuiper Belt.     

Giant planet migration through the action of disk torques and planet-planet scattering

This paper presents a parametric study of giant planet migration through the combined action of disk torques and planet-planet scattering. The torques exerted on planets during Type II migration in circumstellar disks readily decrease the semi-major axes a, whereas scattering between planets increases the orbital eccentricities ?. This paper presents a parametric exploration of the possible parameter space for this migration scenario using two (initial) planetary mass distributions and a range of values for the time scale of eccentricity damping (due to the disk). For each class of systems, many realizations of the simulations are performed in order to determine the distributions of the resulting orbital elements of the surviving planets; this paper presents the results of ∼8500 numerical experiments. Our goal is to study the physics of this particular migration mechanism and to test it against observations of extrasolar planets. The action of disk torques and planet-planet scattering results in a distribution of final orbital elements that fills the a-? plane, in rough agreement with the orbital elements of observed extrasolar planets. In addition to specifying the orbital elements, we characterize this migration mechanism by finding the percentages of ejected and accreted planets, the number of collisions, the dependence of outcomes on planetary masses, the time spent in 2:1 and 3:1 resonances, and the effects of the planetary IMF. We also determine the distribution of inclination angles of surviving planets and the distribution of ejection speeds for exiled planets.     

Orbital stability of systems of closely-spaced planets

An investigation of the stability of systems of 1 M_⊕ (Earth-mass) bodies orbiting a Sun-like star has been conducted for virtual times reaching 10 billion years. For the majority of the tests, a symplectic integrator with a fixed timestep of between 1 and 10 days was employed; however, smaller timesteps and a Bulirsch-Stoer integrator were also selectively utilized to increase confidence in the results. In most cases, the planets were started on initially coplanar, circular orbits, and the longitudinal initial positions of neighboring planets were widely separated. The ratio of the semimajor axes of consecutive planets in each system was approximately uniform (so the spacing between consecutive planets increased slowly in terms of distance from the star). The stability time for a system was taken to be the time at which the orbits of two or more planets crossed. Our results show that, for a given class of system (e.g., three 1 M_⊕ planets), orbit crossing times vary with planetary spacing approximately as a power law over a wide range of separation in semimajor axis. Chaos tests indicate that deviations from this power law persist for changed initial longitudes and also for small but non-trivial changes in orbital spacing. We find that the stability time increases more rapidly at large initial orbital separations than the power-law dependence predicted from moderate initial orbital separations. Systems of five planets are less stable than systems of three planets for a specified semimajor axis spacing. Furthermore, systems of less massive planets can be packed more closely, being about as stable as 1 M_⊕ planets when the radial separation between planets is scaled using the mutual Hill radius. Finally, systems with retrograde planets can be packed substantially more closely than prograde systems with equal numbers of planets.     

Eccentricities of Planets in Binary Systems

The most puzzling property of the extrasolar planets discovered by recent radial velocity surveys is their high orbital eccentricities, which are very difficult to explain within our current theoretical paradigm for planet formation. Current data reveal that at least 25% of these planets, including some with particularly high eccentricities, are orbiting a component of a binary star system. The presence of a distant companion can cause significant secular perturbations in the orbit of a planet. At high relative inclinations, large-amplitude, periodic eccentricity perturbations can occur. These are known as “Kozai cycles” and their amplitude is purely dependent on the relative orbital inclination. Assuming that every planet host star also has a (possibly unseen, e.g., substellar) distant companion, with reasonable distributions of orbital parameters and masses, we determine the resulting eccentricity distribution of planets and compare it to observations? We find that perturbations from a binary companion always appear to produce an excess of planets with both very high (?0.6) and very low (e ? 0.1) eccentricities. The paucity of near-circular orbits in the observed sample implies that at least one additional mechanism must be increasing eccentricities. On the other hand, the overproduction of very high eccentricities observed in our models could be combined with plausible circularization mechanisms (e.g., friction from residual gas) to create more planets with intermediate eccentricities (e? 0.1–0.6).     

On the Inclination Distribution of the Jovian Irregular Satellites

Irregular satellites—moons that occupy large orbits of significant eccentricity e and/or inclination I—circle each of the giant planets. The irregulars often extend close to the orbital stability limit, about 1/3-1/2 of the way to the edge of their planet's Hill sphere. The distant, elongated, and inclined orbits suggest capture, which presumably would give a random distribution of inclinations. Yet, no known irregulars have inclinations (relative to the ecliptic) between 47 and 141°.This paper shows that many high-I orbits are unstable due to secular solar perturbations. High-inclination orbits suffer appreciable periodic changes in eccentricity; large eccentricities can either drive particles with ∼70°<I<110° deep into the realm of the regular satellites (where collisions and scatterings are likely to remove them from planetocentric orbits on a timescale of 10⁷-10⁹ years) or expel them from the Hill sphere of the planet.By carrying out long-term (10⁹ years) orbital integrations for a variety of hypothetical satellites, we demonstrate that solar and planetary perturbations, by causing particles to strike (or to escape) their planet, considerably broaden this zone of avoidance. It grows to at least 55°<I<130° for orbits whose pericenters freely oscillate from 0 to 360°, while particles whose pericenters are locked at ±90° (Kozai mechanism) can remain for longer times.We estimate that the stable phase space (over 10 Myr) for satellites trapped in the Kozai resonance contains ∼10% of all stable orbits, suggesting the possible existence of a family of undiscovered objects at higher inclinations than those currently known.     

From planetesimals to terrestrial planets: N-body simulations including the effects of nebular gas and giant planets

We present results from a suite of N-body simulations that follow the formation and accretion history of the terrestrial planets using a new parallel treecode that we have developed. We initially place 2000 equal size planetesimals between 0.5 and 4.0 AU and the collisional growth is followed until the completion of planetary accretion (>100 Myr). A total of 64 simulations were carried out to explore sensitivity to the key parameters and initial conditions. All the important effect of gas in laminar disks are taken into account: the aerodynamic gas drag, the disk-planet interaction including Type I migration, and the global disk potential which causes inward migration of secular resonances as the gas dissipates. We vary the initial total mass and spatial distribution of the planetesimals, the time scale of dissipation of nebular gas (which dissipates uniformly in space and exponentially in time), and orbits of Jupiter and Saturn. We end up with 1-5 planets in the terrestrial region. In order to maintain sufficient mass in this region in the presence of Type I migration, the time scale of gas dissipation needs to be 1-2 Myr. The final configurations and collisional histories strongly depend on the orbital eccentricity of Jupiter. If today’s eccentricity of Jupiter is used, then most of bodies in the asteroidal region are swept up within the terrestrial region owing to the inward migration of the secular resonance, and giant impacts between protoplanets occur most commonly around 10 Myr. If the orbital eccentricity of Jupiter is close to zero, as suggested in the Nice model, the effect of the secular resonance is negligible and a large amount of mass stays for a long period of time in the asteroidal region. With a circular orbit for Jupiter, giant impacts usually occur around 100 Myr, consistent with the accretion time scale indicated from isotope records. However, we inevitably have an Earth size planet at around 2 AU in this case. It is very difficult to obtain spatially concentrated terrestrial planets together with very late giant impacts, as long as we include all the above effects of gas and assume initial disks similar to the minimum mass solar nebular.     

Orbital stability of systems of closely-spaced planets,II: configurations with coorbital planets

We numerically investigate the stability of systems of 1 \({{\rm M}_{\oplus}}\) planets orbiting a solar-mass star. The systems studied have either 2 or 42 planets per occupied semimajor axis, for a total of 6, 10, 126, or 210 planets, and the planets were started on coplanar, circular orbits with the semimajor axes of the innermost planets at 1 AU. For systems with two planets per occupied orbit, the longitudinal initial locations of planets on a given orbit were separated by either 60° (Trojan planets) or 180°. With 42 planets per semimajor axis, initial longitudes were uniformly spaced. The ratio of the semimajor axes of consecutive coorbital groups in each system was approximately uniform. The instability time for a system was taken to be the first time at which the orbits of two planets with different initial orbital distances crossed. Simulations spanned virtual times of up to 1 × 10⁸, 5 × 10⁵, and 2 × 10⁵ years for the 6- and 10-planet, 126-planet, and 210-planet systems, respectively. Our results show that, for a given class of system (e.g., five pairs of Trojan planets orbiting in the same direction), the relationship between orbit crossing times and planetary spacing is well fit by the functional form log(t _c /t ₀) = b β + c, where t _c is the crossing time, t ₀ = 1 year, β is the separation in initial orbital semimajor axis (in terms of the mutual Hill radii of the planets), and b and c are fitting constants. The same functional form was observed in the previous studies of single planets on nested orbits (Smith and Lissauer 2009). Pairs of Trojan planets are more stable than pairs initially separated by 180°. Systems with retrograde planets (i.e., some planets orbiting in the opposite sense from others) can be packed substantially more closely than can systems with all planets orbiting in the same sense. To have the same characteristic lifetime, systems with 2 or 42 planets per orbit typically need to have about 1.5 or 2 times the orbital separation as orbits occupied by single planets, respectively.     

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.     

Dynamical evolutuion of planetesimals driven by a massive planet: First simulations

N-body simulations of the dynamical evolution of proto-planets embedded in a swarm of planetesimals and perturbed by a massive Jupiter-like planet were performed with the GRAPE-4 system recently installed in Marseille observatory. Initially both the protoplanets and the planetesimals are on circular and coplanar orbits distributed in a ring located within the orbital radius of the perturber. The first simulations show that, for a perturber significantly more massive than Jupiter, the system of the proto-planets becomes strongly unstable with the possibility of orbit crossing.     

Complex of Meteoroid Orbits with Eccentricities Near 1 and Higher

In our work, the method that can help to predict the existence of distant objects in the Solar system is demonstrated. This method is connected with statistical properties of a heliocentric orbital complex of meteoroids with high eccentricities. Heliocentric meteoroid orbits with high eccentricities are escape routes for dust material from distant parental objects with near-circular orbits to Earth-crossing orbits. Ground-based meteor observations yield trajectory information from which we can derive their place of possible origin: comets, asteroids, and other objects (e.g. Kuiper Objects) in the Solar system or even interstellar space. Statistical distributions of radius vectors of nodes, and other parameters of orbits of meteoroids contain key information about position of greater bodies. We analyze meteor orbits with high eccentricities that were registered in 1975–1976 in Kharkiv (Ukraine). The orbital data of the Kharkiv electronic catalogue are received from observations of radiometeors with masses 10⁻⁶−10⁻³ g.