Fractionation of noble gases by thermal escape from accreting planetesimals

A model for the selective loss of noble gases by thermal escape of the gases from planetesimals as they grow to form the terrestrial planets has been developed. The initial elemental and isotopic abundance ratios are assumed to be solar. Competition between gravitational binding and escape determines the degree of fractionation that occurs. Two classes of planetesimals can be formed on a time scale consistent with modern models of accretion. One class is depleted in neon and, in some cases, partly in ³⁶Ar. The other class is neon rich. Subject to the validity of some assumptions regarding loss of planetary atmospheres following collisions between very large embryo planets and a strong radial dependence in the rate of accumulation of neon-rich planetesimals, the mechanism can account for all known properties of the noble gas volatiles on the terrestrial planets except one. This is the ³⁶Ar/³⁸Ar ratios for Earth and Mars which are predicted to be much lower than observed. This failure is probably fatal for the hypothesis.     

Model consideration of the bombardment event of the asteroidal belt by the planetesimals scattered from the Jupiter zone

The temporal evolutions of the planetesimals scattered from the Jupiter zone for different masses of the proto-Jupiter [(a) 0.1 and (b) 1.0 of the present mass] are investigated. Due to the combined effects of the orbital evolution of the planetesimals and the elimination of these projectiles either via impact capture or injection into escape velocity by the outer planets, the whole scattering process lasts about 10⁸ yr for case (a) and about 10⁷ yr for case (b). The longer time scale may be a good estimate for the accretion time interval of Jupiter while the shorter one (10⁷) gives the upper time limit of the late heavy-bombardment epoch of the terrestrial planets due to planetesimals scattered from the Jupiter zone. The limiting value of the encounter velocity U at the end of the scattering process is ≈0.6. Consideration of the collisional interaction of these projectiles with the asteroids indicates that the corresponding bombardment effect could be rather appreciable. Also, the asteroids on the inner edge of the main asteroid belt would have been bombarded more severely than those on the outer edge. From this point of view, the structure of the asteroidal belt could be affected significantly not only by Jupiter's gravitational perturbation effect but also by its early scattering process.     

Growth of large,late-stage planetesimals

The late stage of terrestrial planets' growth determined many of their fundamental properties, including their thermal properties and petrology, their impact records, and possibly the existence of the Moon. A critical result of late-stage models, which bears on observable properties, is the size of the largest planetesimals that grew near, and later impacted,those that became full-size planets. There has been considerable misinterpretation of previous models regarding the relation between the size of planetesimals and their relative velocities. Furthermore, some models neglect the possible decrease in relative velocity as control is transferred from the largest to the second-largest body in an accreation zone. Evidence that Venus helped stir Earth-zone planetismals is not copelling. When models are evaluated, the results are found to depend strongly on uncertain initial conditions. The size of the second-largest planetesimal in the Earth's zone might range from ～300 to ～2500 km, with corresponding accretion times of ～7 × 10⁶ and ～10⁸ years, respectively. Both extremes are generated from plausible initial conditions and both seem consistent with observed planetary properties.     

Planet formation: Compositional mixing and lunar compositional anomalies

Significant fractions of each planet's late-accreted mass originated not at its own distance from the Sun, but from a neighboring planet's orbit, according to results that follow from calculations by Wetherill (1975). “Late-accreted” refers to a loosely defined period after planets acquired most of their present mass. In an idealized model, Mercury, Venus, Earth, and Mars received 47, 45, 37, and 52% of their late-accreted mass from planetesimals formed closer to other planets. Resulting compositional anomalies in outer parts of early planets could be significant; atmospheric tests of Lewis's predicted S deficiency on Venus may be inconclusive.The Moon's orbit around Earth puts it in a special category: sorting occurs between Moon-impacting and Earth-impacting material according to approach velocity. In the above model, the moon receives 60% of its late-accreted mass from planetesimals formed near Venus' orbit. Distant planetesimals could be perturbed into the Earth-Moon system and cause major changes in the Moon's composition with only minor effect on Earth. The entire lunar bulk composition anomaly could be explained by plausible reservoirs of distant low-density material.     

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.     

Mass loss from the region of Mars and the asteroid belt

Models of the solar nebula suggest that the mass of solid matter which condensed in the region of Mars and the asteroids was much greater than the amount now present. Bombardment by a primordial population of asteroidal bodies originating near Jupiter's orbit could preferentially remove matter from this region, without significant effects in the Earth's zone. A “critical velocity” exists, for which they can be ejected from the solar system by Jupiter. The minimum perihelion attainable at this velocity lies between the orbits of Mars and the Earth. The lifetimes of Mars-crossing bodies are limited by collisions with Jupiter; Earth-crossers are ejected on a much shorter time scale. The total bombardment flux was at least two orders of magnitude greater in the zone of Mars than in that of the Earth. The flux at Venus and Mercury from this source was negligible. The cratering rate for Mars may have differed greatly from those of the other terrestrial planets for a significant fraction of the age of the solar system.     

Late stages of planetary accretion

We derive first-order differential equations for the late stages of planetary accretion (planetesimal mass >10¹³ g). The effect of gravitational encounters, energy exchange, collisions, and gas drag has been included. Two simple models are discussed, namely, (i) when all planetesimals have the same mass and (ii) when there is one large planetesimal and numerous small planetesmals. Gravitational two-body encounters are modeled according to Chandrasekhar's classical theory from stellar dynamics. It is shown that the velocity increase due to mutual encounters can be modeled according to the simple theory of random flights. We find analytical equations for the average velocity decrease due to collisions. Gas drag, if present, is modeled in averaged form up to the first order in the eccentricities and inclinations of the planetesimals. Characteristic time scales for the formation of the terrestrial planets are found for the most favorable models to be of order 10⁸ year. The calculated mass of rock and ice of the giant planets is too low as compared to the observed one. This difficulty of our model could be overcome by assuming a several times larger surface density, an enlarged accretion cross section, and gas accretion during the final stages of accretion of the solid cores of the giant planets. Analytical and numerical results are presebted, the evolutionary tracks showing satisfactory agreement with observations for some models.     

Did 26Al and impact‐induced heating differentiate Mercury?

Numerical models dealing with the planetary scale differentiation of Mercury are presented with the short‐lived nuclide, ²⁶Al, as the major heat source along with the impact‐induced heating during the accretion of planets. These two heat sources are considered to have caused differentiation of Mars, a planet with size comparable to Mercury. The chronological records and the thermal modeling of Mars indicate an early differentiation during the initial ~1 million years (Ma) of the formation of the solar system. We theorize that in case Mercury also accreted over an identical time scale, the two heat sources could have differentiated the planets. Although unlike Mars there is no chronological record of Mercury's differentiation, the proposed mechanism is worth investigation. We demonstrate distinct viable scenarios for a wide range of planetary compositions that could have produced the internal structure of Mercury as deduced by the MESSENGER mission, with a metallic iron (Fe‐Ni‐FeS) core of radius ~2000 km and a silicate mantle thickness of ~400 km. The initial compositions were derived from the enstatite and CB (Bencubbin) chondrites that were formed in the reducing environments of the early solar system. We have also considered distinct planetary accretion scenarios to understand their influence on thermal processing. The majority of our models would require impact‐induced mantle stripping of Mercury by hit and run mechanism with a protoplanet subsequent to its differentiation in order to produce the right size of mantle. However, this can be avoided if we increase the Fe‐Ni‐FeS contents to ~71% by weight. Finally, the models presented here can be used to understand the differentiation of Mercury‐like exoplanets and the planetary embryos of Venus and Earth.     

On the process of accretion in the formation of the planets and comets

The physically reasonable assumption that the seed bodies which initiated the accretion of the individual asteroids, planets, and comets (subsequently these objects are collectively called planetoids) formed by stochastic processes requires a radius distribution function which is unique except for two scaling parameters: the total number of planetoids and their most probable radius. The former depends on the ease of formation of the seed bodies while the second is uniquely determined by the average pre-encounter velocity, V, of the accretable material relative to an individual planetoid. This theoretical radius function can be fit to the initial asteroid radius distribution which Anders (1965) derived from the present-day distribution by allowing for fragmentation collisions among the asteroids since their formation. Normalizing the theoretical function to this empirical distribution reveals that there were about 10² precollision asteroids and that V = (2?4) × 10^?2 km/sec which was presumably the turbulent velocity in the Solar Nebula. Knowing V we can determine the scale height of the dust in the Solar Nebula and consequently its space density. The density of accretable material determines the rate of accretion of the planetoids. From this we find, for example, that the Earth formed in about 8 × 10⁶ yr and it attained a maximum temperature through accretion of about 3 × 10³°K. From the total mass of the terrestrial planets and the theoretical radius function we find that about 2 × 10³ planetoids formed in the vicinity of the terrestrial planets. Except for the asteroids the smaller planetoids have since been accreted by the terrestrial planets. About 15% of the present mass of the terrestrial planets was accumulated by the secondary accretion of these smaller primary planetoids. There are far fewer primary planetoids than craters on the Moon or Mars. The craters were likely produced by the collisional breakup of a few primary planetoids with masses between one-tenth and one lunar mass. This deduction comes from comparing the collision cross sections of the planetoids in this mass range to that of the terrestrial planets. This comparison shows that two to three collisions leading to the breakup of four to six objects likely occurred among these objects before their accretion by the terrestrial planets. The number of these fragments is quite adequate to explain the lunar and Martin craters. Furthermore the mass spectrum of such fragments is a power-law distribution which results in a power-law distribution of crater radii of just the type observed on the Moon and Mars. Applying the same analysis to the planetoids which formed in the vicinity of the giant planets reveals that it is unlikely that any fragmentation collisions took place among them before they were accreted by these planets due to the integrated collision cross section of the giant planets being about three orders of magnitude greater than that of the terrestrial planets. We can thus anticipate a marked scarcity of impact craters on the satellites of these outer planets. This prediction can be tested by future space probes. Our knowledge of the radius function of the comets is consistent with their being primary planetoids. The primary difference between the radius function of the planetoids which formed in the inner part of the solar system and that of the comets results from the fact that the seed bodies which grew into the comets formed far more easily than those which grew into the asteroids and the terrestrial planets. Thus in the outer part of the Solar Nebula the principal solid material (water and ammonia snow) accreted into a huge (～10¹²⁺) number of relatively small objects (comets) while in the inner part of the nebula the solid material (hard-to-stick refractory substances) accumulated into only a few (～10³) large objects (asteroids and terrestrial planets). Uranus and Neptune presumably formed by the secondary accretion of the comets.     

Characterization of terrestrial exoplanets based on the phase curves and albedos of Mercury, Venus and Mars

The empirically derived phase curves of terrestrial planets strongly distinguish between airless Mercury, cloud-covered Venus, and the intermediate case of Mars. The function for Mercury is steeply peaked near phase angle zero due to powerful backscattering from its surface, while that for Venus has 100 times less contrast and exhibits a brightness excess near 170° due to Mie scattering from droplets in the atmosphere. The phase curve of Mars falls between those of Mercury and Venus, and there are variations in luminosity due to the planet’s rotation, seasons, and atmospheric states. The phase function and geometric albedo of the Earth are estimated from published albedos values. The curves for Mercury, Venus and Mars are compared to that of the Earth as well as theoretical phase functions for giant planets. The parameters of these different phase functions can be used to characterize exoplanets.     

Accretion in the inner nebula: The relationship between terrestrial planetary compositions and meteorites*

Abstract— The bulk compositions of the terrestrial planets are assessed. Venus and Earth probably have similar bulk compositions, but Mars is enriched in volatile elements. The inner planets are all depleted in volatile elements, as shown by K/U ratios, relative to most meteorites and the CI primordial values. Terrestrial upper mantle Mg/Si ratios are high compared with CI data. If they are representative of the bulk Earth, then the Earth accreted from a segregated suite of planetesimals that had non-chondritic major element abundances. The CI meteorite abundances, despite aqueous alteration, match the solar data and provide the best estimate for the composition of the solar nebula, including the iron abundance. The widespread depletion of volatile elements in the inner solar nebula is most likely caused by heating related to early violent solar activity (e.g., T Tauri and FU Orionis stages) which, for example, drove water out to a “snow line” in the vicinity of Jupiter. The variation in composition among the meteorites and the apparent lack of mixing among the groups indicates accretion from narrow feeding zones. There appears to have been little mixing between meteorite and planetary formation zones, as shown by the oxygen isotope variations, lack of mixing of meteorite groups, and differences in K/U ratios. In summary, it appears that the final accretion of planets did not result in widespread homogenization, and that mixing zones were not more than about 0.3 A.U. wide. Although the composition of the Moon is unique, and its origin due to an essentially random event, its presence reinforces the planetesimal hypothesis and the importance of stochastic processes during planetary accretion in the inner solar system.     

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.     

On the internal structures of mercury and venus

Recent radar measures of the radius and mass of Mercury imply a composition for the planet containing about 60% iron. One or other of two conclusions seems inescapable: either that Mercury is a highly exceptional object among terrestrial planets, or that all measures to date of the planet involve substantial systematic error. In either case the situation is such that independent checking of the radius and mass of Mercury by some entirely different means has become of the greatest importance to planetary physics and cosmogony.The recent radar and other determinations of the solid radius of Venus imply an internal structure similar to that of the Earth, namely a liquid core surrounded by a solid mantle and outer-shell zone. The theory also implies that the temperatures within Venus should be slightly higher than at the corresponding parts of the Earth. The proportion of mass in the core of Venus (about 25% of the whole) is entirely consistent with the phase-change hypothesis as to its nature, as of course is also the absence of any liquid or iron core in both Mars and the Moon. On the older iron-core hypothesis, Venus with considerably less iron content by mass than the Earth, and Mars and the Moon with none, would all present problems in different degrees to account for the differences of composition.If Venus began as an all-solid planet, the initial radius would have been about 6300 km, and the total amount of surface reduction to date owing to contraction of the planet would have been almost 40 million km², and as a proportion of the total area only slightly less than the contraction of the Earth. The theory thus predicts the existence of folded and thrusted mountain-systems of terrestrial type at the surface of Venus.     

Gaseous drag and planetary formation by accretion

We have made numerical experiments of the collisional and gravitational interaction of a planetesimal swarm in the early Solar System. In particular we study the dynamical evolution of an initial population of kilometer-size planetesimals subject to collisions (accretion, rebound, cratering, and catastrophic fragmentation). This study is based on a Monte-Carlo statistical method and provides the mass and velocity distributions of the planetesimal swarm as a function of time as well as their distribution in heliocentric distance. Several experiments have been performed and three of them are presented here. They simulate the accretional growth of numerous planetesimals in the absence (or presence) of gaseous drag, with (or without) one larger embryo among them, and with (or without) a size gradient. The results show that (i) for a population of planetesimals submitted to a negative gradient in size as the heliocentric distance increases, the outer planetesimals spiral toward the Sun faster than inner ones, leading after some time to an accumulation of bodies inside the cloud which allows the formation of an embryo; (ii) the growth of one embryo among a population of planetesimals is accelerated by the presence of gas and is warranted as long as its feeding zone is fed by the inward flow of planetesimals due to gas drag. These results offer some complementary new insights in the understanding of the accretional formation of 4–5 terrestrial planets instead of the numerous Moon-size planets generally found in numerical experiments.     

Relative velocities of planetesimals and the early accumulation of planets

Safronov's statement that relative velocities of planetesimals are on the order of the escape velocity of the largest body of the population is shown to be correct only when a major part of the total mass resides in several large bodies. In the first stage of accumulation, runaway accretion produces large bodies separated by mass form the remaining population. At this stage, relative velocities of planetesimals are much smaller than those adopted earlier. This requires a modification of Schmidt's scheme of accumulation of the Earth and other terrestrial planets from material in their feeding zones. This also leads to removal of the author's arguments (Levin 1972c) in favor of a protoplanetary nebula with an extended, massive periphery.Paper presented at the Conference on Protostars and Planets, held at the Planetary Science Institute, University of Arizona, Tucson, Arizona, between January 3 and 7, 1978.     

The Leonard Award Address Presented 1998 July 27, Dublin,Ireland: On the difficulties of making Earth-like planets

Abstract— Here I discuss the series of events that led to the formation and evolution of our planet to examine why the Earth is unique in the solar system. A multitude of factors are involved: These begin with the initial size and angular momentum of the fragment that separated from a molecular cloud; such random factors are crucial in determining whether a planetary system or a double star develops from the resulting nebula. Another requirement is that there must be an adequate concentration of heavy elements to provide the 2% “rock” and “ice” components of the original nebula. An essential step in forming rocky planets in the inner nebula is the loss of gas and depletion of volatile elements, due to early solar activity that is linked to the mass of the central star. The lifetime of the gaseous nebula controls the formation of gas giants. In our system, fine timing was needed to form the gas giant, Jupiter, before the gas in the nebula was depleted. Although Uranus and Neptune eventually formed cores large enough to capture gas, they missed out and ended as ice giants. The early formation of Jupiter is responsible for the existence of the asteroid belt (and our supply of meteorites) and the small size of Mars, whereas the gas giant now acts as a gravitational shield for the terrestrial planets. The Earth and the other inner planets accreted long after the giant planets, from volatile-depleted planetesimals that were probably already differentiated into metallic cores and silicate mantles in a gas-free, inner nebula. The accumulation of the Earth from such planetesimals was essentially a stochastic process, accounting for the differences among the four rocky inner planets—including the startling contrast between those two apparent twins, Earth and Venus. Impact history and accretion of a few more or less planetesimals were apparently crucial. The origin of the Moon by a single massive impact with a body larger than Mars accounts for the obliquity (and its stability) and spin of the Earth, in addition to explaining the angular momentum, orbital characteristics, and unique composition of the Moon. Plate tectonics (unique among the terrestrial planets) led to the development of the continental crust on the Earth, an essential platform for the evolution of Homo sapiens. Random major impacts have punctuated the geological record, accentuating the directionless course of evolution. Thus a massive asteroidal impact terminated the Cretaceous Period, resulted in the extinction of at least 70% of species living at that time, and led to the rise of mammals. This sequence of events that resulted in the formation and evolution of our planet were thus unique within our system. The individual nature of the eight planets is repeated among the 60-odd satellites—no two appear identical. This survey of our solar system raises the question whether the random sequence of events that led to the formation of the Earth are likely to be repeated in detail elsewhere. Preliminary evidence from the “new planets” is not reassuring. The discovery of other planetary systems has removed the previous belief that they would consist of a central star surrounded by an inner zone of rocky planets and an outer zone of giant planets beyond a few astronomical units (AU). Jupiter-sized bodies in close orbits around other stars probably formed in a similar manner to our giant planets at several astronomical units from their parent star and, subsequently, migrated inwards becoming stranded in close but stable orbits as “hot Jupiters”, when the nebula gas was depleted. Such events would prevent the formation of terrestrial-type planets in such systems.     

A co-accretional model of satellite formation

Numerical calculation of a simple accretion model including the effects of tidal friction indicate that coformation is tenable only if the planet's Q is less than about 10³. The parameter which most strongly affects the final mass ratio of the pair is the time at which the secondary embryo is introduced. Our model yields the proper Moon-Earth mass ratio if the Moon embryo is introduced when the Earth is only about $\text{1}\text{10}$ of its final mass. The lunar orbit remains at about 10 Earth radii throughout most of the growth.This model of satellite formation overcomes two difficulties of the “circumterrestrial cloud” model of Ruskol (1960, 1963, 1972): (1) The difficulty of accumulating a mass as great as the entire Moon before gravitational instability reduces the cloud to a small number of moonlets is removed. (2) The differences between terrestrial and outer planet satellite systems is easily understood in terms of the differences in Q between these planets. The high Q of the outer planets does not allow a satellite embryo to survive a significant portion of the accretion process, thus only small bodies which formed very late in the accumulation of the planet remain as satellites. The low Q of the terrestrial planets allows satellite embryos of these planets to survive during accretion, thus massive satellites such as the Earth's Moon are expected. The present lack of such satellites of the other terrestrial planets may be the result of tidal evolution, either infall following primary despinning (Burns, 1973) or escape due to increase in orbit eccentricity.     

Formation of the terrestrial planets

The early phases of formation in the inner solar system were dominated by collisions and short-range dynamical interactions among planetesimals. But the later phases, which account for most of the differences among planets, are unsure because the dynamics are more subtle. Jupiter's influence became more important, leading to drastic clearing out of the asteroid belt and the stunting of Mars's growth. Further in, the effect of Jupiter-- both directly and indirectly, through ejection of mass in the outer solar system-- was probably to speed up the process without greatly affecting the outcome. The great variety in bulk properties of the terrestrial bodies indicate a terminal phase of great collisions, so that the outcome is the result of small-N statistics. Mercury, 65 percent iron, appears to be a residual core from a high-velocity collision. All planets appear to require a late phase of high energy impacts to erode their atmospheres: including the Earth, to remove CO₂ so that its ocean could form by condensation of water.Consistent with this model is that the largest collision, about 0.2 Earth masses, was into the proto-Earth, although the only property that appears to require it is the great lack of iron in the Moon. The other large differences between the Earth and Venus, angular momentum (spin plus satellite) and inert gas abundances, must arise from origin circumstances, but neither require nor forbid the giant impact. Venus's higher ratio of light to heavy inert gases argues for it receiving a large icy impactor, about 10^–6 Earth masses from far out, requiring some improbable dynamics to get a low enough approach velocity. Core formation in both planets probably started rather early during accretion.Some geochemical evidences argue for the Moon coming from the Earth's mantle, but are inconclusive. Large scale melting of the mantle by the giant impact would plausibly have led to stratification. But the "lock-up" at the end of turbulent mantle convection is a trade-off between rates: crystallization of constituents of small density difference versus overall freezing. Also, factors such as differences in melting temperatures and densities, melt compressibilities, and phase transitions may have had homogenizing effects in the subsequent mantle convection.     

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.     

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.