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_⊕.     

The Effect of Tidal Interaction with a Gas Disk on Formation of Terrestrial Planets

We have performed N-body simulation on final accretion stage of terrestrial planets, including the effect of damping of eccentricity and inclination caused by tidal interaction with a remnant gas disk. As a result of runway and oligarchic accretion, about 20 Mars-sized protoplanets would be formed in nearly circular orbits with orbital separation of several to ten Hill radius. The orbits of the protoplanets would be eventually destabilized by long-term mutual gravity and/or secular resonance of giant gaseous planets. The protoplanets would coalesce with each other to form terrestrial planets through the orbital crossing. Previous N-body simulations, however, showed that the final eccentricities of planets are around 0.1, which are about 10 times higher than the present eccentricities of Earth and Venus. The obtained high eccentricities are the remnant of orbital crossing. We included the effect of eccentricity damping caused by gravitational interaction with disk gas as a drag force (“gravitational drag”) and carried out N-body simulation of accretion of protoplanets. We start with 15 protoplanets with 0.2M⊕ and integrate the orbits for 10⁷ years, which is consistent with the observationally inferred disk lifetime (in some runs, we start with 30 protoplanets with 0.1M⊕). In most runs, the damping time scale, which is equivalent to the strength of the drag force, is kept constant throughout each run in order to clarify the effects of the damping. We found that the planets' final mass, spatial distribution, and eccentricities depend on the damping time scale. If the damping time scale for a 0.2M⊕ mass planet at 1 AU is longer than 10⁸ years, planets grow to Earth's size, but the final eccentricities are too high as in gas-free cases. If it is shorter than 10⁶ years, the eccentricities of the protoplanets cannot be pumped up, resulting in not enough orbital crossing to make Earth-sized planets. Small planets with low eccentricities are formed with small orbital separation. On the other hand, if it is between 10⁶ and 10⁸ years, which may correspond to a mostly depleted disk (0.01-0.1% of surface density of the minimum mass model), some protoplanets can grow to about the size of Earth and Venus, and the eccentricities of such surviving planets can be diminished within the disk lifetime. Furthermore, in innermost and outermost regions in the same system, we often find planets with smaller size and larger eccentricities too, which could be analogous to Mars and Mercury. This is partly because the gravitational drag is less effective for smaller mass planets, and partly due to the “edge effect,” which means the innermost and outermost planets tend to remain without collision. We also carried out several runs with time-dependent drag force according to depletion of a gas disk. In these runs, we used exponential decay model with e-folding time of 3×10⁶ years. The orbits of protoplanets are stablized by the eccentricity damping in the early time. When disk surface density decays to ?1% of the minimum mass disk model, the damping force is no longer strong enough to inhibit the increase of the eccentricity by distant perturbations among protoplanets so that the orbital crossing starts. In this disk decay model, a gas disk with 10⁻⁴-10⁻³ times the minimum mass model still remains after the orbital crossing and accretional events, which is enough to damp the eccentricities of the Earth-sized planets to the order of 0.01. Using these results, we discuss a possible scenario for the last stage of terrestrial planet formation.     

High-resolution simulations of the final assembly of Earth-like planets I. Terrestrial accretion and dynamics

The final stage in the formation of terrestrial planets consists of the accumulation of ∼1000-km “planetary embryos” and a swarm of billions of 1-10 km “planetesimals.” During this process, water-rich material is accreted by the terrestrial planets via impacts of water-rich bodies from beyond roughly 2.5 AU. We present results from five high-resolution dynamical simulations. These start from 1000-2000 embryos and planetesimals, roughly 5-10 times more particles than in previous simulations. Each simulation formed 2-4 terrestrial planets with masses between 0.4 and 2.6 Earth masses. The eccentricities of most planets were ∼0.05, lower than in previous simulations, but still higher than for Venus, Earth and Mars. Each planet accreted at least the Earth's current water budget. We demonstrate several new aspects of the accretion process: (1) The feeding zones of terrestrial planets change in time, widening and moving outward. Even in the presence of Jupiter, water-rich material from beyond 2.5 AU is not accreted for several millions of years. (2) Even in the absence of secular resonances, the asteroid belt is cleared of >99% of its original mass by self-scattering of bodies into resonances with Jupiter. (3) If planetary embryos form relatively slowly, then the formation of embryos in the asteroid belt may have been stunted by the presence of Jupiter. (4) Self-interacting planetesimals feel dynamical friction from other small bodies, which has important effects on the eccentricity evolution and outcome of a simulation.     

Terrestrial planet formation surrounding close binary stars

Most stars reside in binary/multiple star systems; however, previous models of planet formation have studied growth of bodies orbiting an isolated single star. Disk material has been observed around both components of some young close binary star systems. Additionally, it has been shown that if planets form at the right places within such disks, they can remain dynamically stable for very long times. Herein, we numerically simulate the late stages of terrestrial planet growth in circumbinary disks around ‘close’ binary star systems with stellar separations 0.05 AU?aB?0.4 AU and binary eccentricities 0?eB?0.8. In each simulation, the sum of the masses of the two stars is 1 M_⊙, and giant planets are included. The initial disk of planetary embryos is the same as that used for simulating the late stages of terrestrial planet formation within our Solar System by Chambers [Chambers, J.E., 2001. Icarus 152, 205-224], and around each individual component of the α Centauri AB binary star system by Quintana et al. [Quintana, E.V., Lissauer, J.J., Chambers, J.E., Duncan, M.J., 2002. Astrophys. J. 576, 982-996]. Multiple simulations are performed for each binary star system under study, and our results are statistically compared to a set of planet formation simulations in the Sun-Jupiter-Saturn system that begin with essentially the same initial disk of protoplanets. The planetary systems formed around binaries with apastron distances QB≡aB(1+eB)?0.2 AU are very similar to those around single stars, whereas those with larger maximum separations tend to be sparcer, with fewer planets, especially interior to 1 AU. We also provide formulae that can be used to scale results of planetary accretion simulations to various systems with different total stellar mass, disk sizes, and planetesimal masses and densities.     

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.     

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.     

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.     

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.     

Building the terrestrial planets: Constrained accretion in the inner Solar System

To date, no accretion model has succeeded in reproducing all observed constraints in the inner Solar System. These constraints include: (1) the orbits, in particular the small eccentricities, and (2) the masses of the terrestrial planets - Mars’ relatively small mass in particular has not been adequately reproduced in previous simulations; (3) the formation timescales of Earth and Mars, as interpreted from Hf/W isotopes; (4) the bulk structure of the asteroid belt, in particular the lack of an imprint of planetary embryo-sized objects; and (5) Earth’s relatively large water content, assuming that it was delivered in the form of water-rich primitive asteroidal material. Here we present results of 40 high-resolution (N = 1000-2000) dynamical simulations of late-stage planetary accretion with the goal of reproducing these constraints, although neglecting the planet Mercury. We assume that Jupiter and Saturn are fully-formed at the start of each simulation, and test orbital configurations that are both consistent with and contrary to the “Nice model”. We find that a configuration with Jupiter and Saturn on circular orbits forms low-eccentricity terrestrial planets and a water-rich Earth on the correct timescale, but Mars’ mass is too large by a factor of 5-10 and embryos are often stranded in the asteroid belt. A configuration with Jupiter and Saturn in their current locations but with slightly higher initial eccentricities (e = 0.07-0.1) produces a small Mars, an embryo-free asteroid belt, and a reasonable Earth analog but rarely allows water delivery to Earth. None of the configurations we tested reproduced all the observed constraints. Our simulations leave us with a problem: we can reasonably satisfy the observed constraints (except for Earth’s water) with a configuration of Jupiter and Saturn that is at best marginally consistent with models of the outer Solar System, as it does not allow for any outer planet migration after a few Myr. Alternately, giant planet configurations which are consistent with the Nice model fail to reproduce Mars’ small size.     

Models of Jupiter and Saturn after Galileo mission

New models of Jupiter are based on observational data provided by the Galileo spaceprobe, which considerably improved previously existing estimates of the helium abundance in the atmosphere of Jupiter. These data yield for Jupiter’s atmosphere 20% of the solar oxygen abundance and do not agree with the results of the analysis of the collision of comet Shoemaker-Levy 9 with Jupiter (10 times the solar value). Therefore, both the models of Jupiter with water-depleted and water-enriched atmosphere are considered. By analogy with Jupiter, trial models of Saturn with a water-depleted external envelope are also developed. The molecular-metallic phase transition pressure of hydrogen P_m was taken to be 1.5, 2 and 3 Mbar. Since Saturn’s internal molecular envelope is noticeably enriched in the IR-component (its weight concentration, 0.25–0.30, being by a factor of 3–4 higher than in Jupiter), the phase transition pressure in Saturn can be lower than in Jupiter. In the constructed models, the IR-core masses are 3–3.5 M_⊕ for Jupiter and 3–5.5 M_⊕ for Saturn. Jupiter’s and Saturn’s IR-cores can be considered embryos onto which the accretion of the gas occurred during the formation of the planets. The mass of the hydrogen–helium component dispersed in the zone of planetary formation constitutes ≈2–5 planetary masses for Jupiter and ≈11–14 planetary masses for Saturn.     

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.     

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.     

Terrestrial planet formation with strong dynamical friction

We have performed 8 numerical simulations of the final stages of accretion of the terrestrial planets, each starting with over 5× more gravitationally interacting bodies than in any previous simulations. We use a bimodal initial population spanning the region from 0.3 to 4 AU with 25 roughly Mars-mass embryos and an equal mass of material in a population of ∼1000 smaller planetesimals, consistent with models of the oligarchic growth of protoplanetary embryos. Given the large number of small planetesimals in our simulations, we are able to more accurately treat the effects of dynamical friction during the accretion process. We find that dynamical friction can significantly lower the timescales for accretion of the terrestrial planets and leads to systems of terrestrial planets that are much less dynamically excited than in previous simulations with fewer initial bodies. In addition, we study the effects of the orbits of Jupiter and Saturn on the final planetary systems by running 4 of our simulations with the present, eccentric orbits of Jupiter and Saturn (the EJS simulations) and the other 4 using a nearly circular and co-planar Jupiter and Saturn as predicted in the Nice Model of the evolution of the outer Solar System [Gomes, R., Levison, H.F., Tsiganis, K., Morbidelli, A., 2005. Nature 435, 466-469; Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461; Morbidelli, A., Levison, H.F., Tsiganis, K., Gomes, R., 2005. Nature 435, 462-465] (the CJS simulations). Our EJS simulations provide a better match to our Solar System in terms of the number and average mass of the final planets and the mass-weighted mean semi-major axis of the final planetary systems, although increased dynamical friction can potentially improve the fit of the CJS simulations as well. However, we find that in our EJS simulations, essentially no water-bearing material from the outer asteroid belt ends up in the final terrestrial planets, while a large amount is delivered in the CJS simulations. In addition, the terrestrial planets in the EJS simulations receive a late veneer of material after the last giant impact event that is likely too massive to reconcile with the siderophile abundances in the Earth's mantle, while the late veneer in the CJS simulations is much more consistent with geochemical evidence.     

Numerical simulation of the final stages of terrestrial planet formation

Three representative numerical simulations of the growth of the terrestrial planets by accretion of large protoplanets are presented. The mass and relative-velocity distributions of the bodies in these simulations are free to evolve simultaneously in response to close gravitational encounters and occasional collisions between bodies. The collisions between bodies, therefore, arise in a natural way and the assumption of expressions for the relative velocity distribution and the gravitational collision cross section is unnecessary. These simulations indicate that the growth of bodies with final masses approaching those of Venus and the Earth is possible, at least for the case of a two-dimensional system. Simulations assuming an initial uniform distribution of orbital eccentricities on the interval from 0 to e_max are found to produce final states containing too many bodies with masses which are too small when e_max < 0.10, while simulations with e_max > 0.20 result in too many catastrophic collisions between bodies thus preventing rapid accretion of planetary-size bodies. The e_max = 0.15 simulation ends with a state surprisingly similar to that of the present terrestrial planets and, therefore, provides a rough estimate of the range of radial sampling to be expected for the terrestrial planets.     

Accretion of the gaseous envelope of Jupiter around a 5-10 Earth-mass core

New numerical simulations of the formation and evolution of Jupiter are presented. The formation model assumes that first a solid core of several M_⊕ accretes from the planetesimals in the protoplanetary disk, and then the core captures a massive gaseous envelope from the protoplanetary disk. Earlier studies of the core accretion-gas capture model [Pollack, J.B., Hubickyj, O., Bodenheimer, P., Lissauer, J.J., Podolak, M., Greenzweig, Y., 1996. Icarus 124, 62-85] demonstrated that it was possible for Jupiter to accrete with a solid core of 10-30 M_⊕ in a total formation time comparable to the observed lifetime of protoplanetary disks. Recent interior models of Jupiter and Saturn that agree with all observational constraints suggest that Jupiter's core mass is 0-11 M_⊕ and Saturn's is 9-22 M_⊕ [Saumon, G., Guillot, T., 2004. Astrophys. J. 609, 1170-1180]. We have computed simulations of the growth of Jupiter using various values for the opacity produced by grains in the protoplanet's atmosphere and for the initial planetesimal surface density, σinit,Z, in the protoplanetary disk. We also explore the implications of halting the solid accretion at selected core mass values during the protoplanet's growth. Halting planetesimal accretion at low core mass simulates the presence of a competing embryo, and decreasing the atmospheric opacity due to grains emulates the settling and coagulation of grains within the protoplanet's atmosphere. We examine the effects of adjusting these parameters to determine whether or not gas runaway can occur for small mass cores on a reasonable timescale. We compute four series of simulations with the latest version of our code, which contains updated equation of state and opacity tables as well as other improvements. Each series consists of a run without a cutoff in planetesimal accretion, plus up to three runs with a cutoff at a particular core mass. The first series of runs is computed with an atmospheric opacity due to grains (hereafter referred to as ‘grain opacity’) that is 2% of the interstellar value and . Cutoff runs are computed for core masses of 10, 5, and 3 M_⊕. The second series of Jupiter models is computed with the grain opacity at the full interstellar value and . Cutoff runs are computed for core masses of 10 and 5 M_⊕. The third series of runs is computed with the grain opacity at 2% of the interstellar value and . One cutoff run is computed with a core mass of 5 M_⊕. The final series consists of one run, without a cutoff, which is computed with a temperature dependent grain opacity (i.e., 2% of the interstellar value for ramping up to the full interstellar value for ) and . Our results demonstrate that reducing grain opacities results in formation times less than half of those for models computed with full interstellar grain opacity values. The reduction of opacity due to grains in the upper portion of the envelope with has the largest effect on the lowering of the formation time. If the accretion of planetesimals is not cut off prior to the accretion of gas, then decreasing the surface density of planetesimals lowers the final core mass of the protoplanet, but increases the formation timescale considerably. Finally, a core mass cutoff results in a reduction of the time needed for a protoplanet to evolve to the stage of runaway gas accretion, provided the cutoff mass is sufficiently large. The overall results indicate that, with reasonable parameters, it is possible that Jupiter formed at 5 AU via the core accretion process in 1 Myr with a core of 10 M_⊕ or in 5 Myr with a core of 5 M_⊕.     

The effects of nebula surface density profile and giant‐planet eccentricities on planetary accretion in the inner solar system

Abstract— We describe results of 32 N‐body planetary accretion simulations that investigate the dependence of terrestrial‐planet formation on nebula surface density profile σ and evolution of the eccentricities of Jupiter and Saturn e_j,s. Two surface density profiles are examined: a decaying profile with σ ∝ 1/a, where a is orbital semi‐major axis, and a peaked profile in which σ increases for a < 2 AU and decreases for a > 2 AU. The peaked profiles are generated by models of coagulation in an initially hot nebula. Models with initial e_j,s = 0.05 (the current value) and 0.1 are considered. Simulations using the decaying profile with e_j,s = 0.1 produce systems most like the observed planets in terms of mass‐weighted mean a and the absence of a planet in the asteroid belt. Simulations with doubled σ produce planets roughly twice as massive as the nominal case. Most initial embryos are removed in each simulation via ejection from the solar system or collision with the Sun. The asteroid belt is almost entirely cleared on a timescale of 10–100 Ma that depends sensitively on e_j,s. Most initial mass with a < 2 AU survives, with the degree of mass loss increasing with a. Mass loss from the terrestrial region occurs on a timescale that is long compared to the mass loss time for the asteroid belt. Substantial radial mixing of material occurs in all simulations, but is greater in simulations with initital e_j,s = 0.05. The degree of mixing is equivalent to a feeding zone of half width 1.5 and 0.9 AU for an Earth mass planet at 1 AU for the cases e_j,s = 0.05 and 0.1, respectively. In simulations with e_j,s = 0.05, roughly one‐third and 5–10% of the mass contained in final terrestrial planets originated in the region a > 2.5 AU for the decaying and peaked profiles, respectively. In the case e_j,s = 0.1, the median mass accreted from a > 2.5 AU is zero for both profiles.     

Could CoRoT-7b and Kepler-10b be remnants of evaporated gas or ice giants?

We present thermal mass loss calculations over evolutionary time scales for the investigation if the smallest transiting rocky exoplanets CoRoT-7b (∼1.68R_Earth) and Kepler-10b (∼1.416R_Earth) could be remnants of an initially more massive hydrogen-rich gas giant or a hot Neptune-class exoplanet. We apply a thermal mass loss formula which yields results that are comparable to hydrodynamic loss models. Our approach considers the effect of the Roche lobe, realistic heating efficiencies and a radius scaling law derived from observations of hot Jupiters. We study the influence of the mean planetary density on the thermal mass loss by placing hypothetical exoplanets with the characteristics of Jupiter, Saturn, Neptune, and Uranus to the orbital location of CoRoT-7b at 0.017 AU and Kepler-10b at 0.01684 AU and assuming that these planets orbit a K- or G-type host star. Our findings indicate that hydrogen-rich gas giants within the mass domain of Saturn or Jupiter cannot thermally lose such an amount of mass that CoRoT-7b and Kepler-10b would result in a rocky residue. Moreover, our calculations show that the present time mass of both rocky exoplanets can be neither a result of evaporation of a hydrogen envelope of a “Hot Neptune” nor a “Hot Uranus”-class object. Depending on the initial density and mass, these planets most likely were always rocky planets which could lose a thin hydrogen envelope, but not cores of thermally evaporated initially much more massive and larger objects.     

Oligarchic growth of giant planets

Runaway growth ends when the largest protoplanets dominate the dynamics of the planetesimal disk; the subsequent self-limiting accretion mode is referred to as “oligarchic growth.” Here, we begin by expanding on the existing analytic model of the oligarchic growth regime. From this, we derive global estimates of the planet formation rate throughout a protoplanetary disk. We find that a relatively high-mass protoplanetary disk (∼10 × minimum-mass) is required to produce giant planet core-sized bodies (∼10 M_⊕) within the lifetime of the nebular gas (?10 million years). However, an implausibly massive disk is needed to produce even an Earth mass at the orbit of Uranus by 10 Myrs. Subsequent accretion without the dissipational effect of gas is even slower and less efficient. In the limit of noninteracting planetesimals, a reasonable-mass disk is unable to produce bodies the size of the Solar System’s two outer giant planets at their current locations on any timescale; if collisional damping of planetesimal random velocities is sufficiently effective, though, it may be possible for a Uranus/Neptune to form in situ in less than the age of the Solar System. We perform numerical simulations of oligarchic growth with gas and find that protoplanet growth rates agree reasonably well with the analytic model as long as protoplanet masses are well below their estimated final masses. However, accretion stalls earlier than predicted, so that the largest final protoplanet masses are smaller than those given by the model. Thus the oligarchic growth model, in the form developed here, appears to provide an upper limit for the efficiency of giant planet formation.     

Dynamical evolution of a cometary swarm in the outer planetary region

The dynamical evolution of bodies under the gravitational influence of the accreting proto-Uranus and proto-Neptune is investigated. The main aim of this study is to analyze the interrelations between the accretion of Uranus and Neptune with other processes of cosmological importance as, for example, the formation of a cometary reservoir from bodies placed into near-parabolic orbits by planetary perturbations and the scattering of bodies to the region of the terrestrial planets. Starting with a mass ratio (initial mass/present mass) of 0.1, Uranus and Neptune acquire masses close to their present ones in a time scale of 10⁸ years. Neptune is found to be the most important contributor of comets to the cometary reservoir. The time scale of bodies scattered by Neptune to reach near-parabolic orbits (semimajor axes a > 10⁴ AU)is about 10⁹ years. The contribution of Uranus was partially inhibited because a large part of the residual bodies of its accretion zone fell under the strong gravitational influence of Jupiter and Saturn. A significant fraction of the bodies dispersed by Uranus and Neptune reached the region of the terrestrial planets in a time scale of some 10⁸ years.     

The effects of metallicity and grain growth and settling on the early evolution of gaseous protoplanets

Giant protoplanets formed by gravitational instability in the outer regions of circumstellar disks go through an early phase of quasi-static contraction during which radii are large (∼1 AU) and internal temperatures are low (<2000 K). The main source of opacity in these objects is dust grains. We investigate two problems involving the effect of opacity on the evolution of isolated, non-accreting planets of 3, 5, and 7 M_J. First, we pick three different overall metallicities for the planet and simply scale the opacity accordingly. We show that higher metallicity results in slower contraction as a result of higher opacity. It is found that the pre-collapse time scale is proportional to the metallicity. In this scenario, survival of giant planets formed by gravitational instability is predicted to be more likely around low-metallicity stars, since they evolve to the point of collapse to small size on shorter time scales. But metal-rich planets, as a result of longer contraction times, have the best opportunity to capture planetesimals and form heavy-element cores. Second, we investigate the effects of opacity reduction as a result of grain growth and settling, for the same three planetary masses and for three different values of overall metallicity. When these processes are included, the pre-collapse time scale is found to be of order 1000 years for the three masses, significantly shorter than the time scale calculated without these effects. In this case the time scale is found to be relatively insensitive to planetary mass and composition. However, the effects of planetary rotation and accretion of gas and dust, which could increase the timescale, are not included in the calculation. The short time scale we find would preclude metal enrichment by planetesimal capture, as well as heavy-element core formation, over a large range of planetary masses and metallicities.