Giant planet formation

The accumulation of giant planets involves processes typical for terrestrial planet formation as well as gasdynamic processes that were previously known only in stars. The condensible element cores of the gas-giants grow by solid body accretion while envelope formation is governed by stellar-like equilibria and the dynamic departures thereof. Two hypotheses for forming Uranus/Neptune-type planets — at sufficiently large heliocentric distances while allowing accretion of massive gaseous envelopes, i.e. Jupiter-type planets at intermediate distances — have been worked out in detailed numerical calculations: (1) Hydrostatic gas-accretion models with time-dependent solid body accretion-rates show a slow-down of core-accretion at the appropriate masses of Uranus and Neptune. As a consequence, gas-accretion also stagnates and a window is opened for removing the solar nebula during a time of roughly constant envelope mass. (2) Gasdynamic calculations of envelope accretion for constant planetesimal accretion-rates show a dynamic transition to new envelope equilibria at the so called critical mass. For a wide range of solar nebula conditions the new envelopes have respective masses similar to those of Uranus and Neptune and are more tightly bound to the cores. The transitions occur under lower density conditions typical for the outer parts of the solar nebula, whereas for higher densities, i.e. closer to the Sun, gasdynamic envelope accretion sets in and is able to proceed to Jupiter-masses.     

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.     

Planetary growth with collisional fragmentation and gas drag

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

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.     

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.     

Hydrodynamic instability of the solar nebula in the presence of a planetary core

When a planetary core composed of condensed matter is accumulated in the primitive solar nebula, the gas of the nebula becomes gravitationally concentrated as an envelope surrounding the planetary core. Models of such gaseous envelopes have been constructed subject to the assumption that the gas everywhere is on the same adiabat as that in the surrounding nebula. The gaseous envelope extends from the surface of the core to the distance at which the gravitational attraction of core plus envelope becomes equal to the gradient of the gravitational potential in the solar nebula; at this point the pressure and temperature of the gas in the envelope are required to attain the background values characteristics of the solar nebula. In general, as the mass of the condensed core increases, increasing amounts of gas became concentrated in the envelope, and these envelopes are stable against hydrodynamic instabilities. However, the core mass then goes through a maximum and starts to decrease. In most of the models tested, the envelopes were hydrodynamically unstable beyond the peak in the core mass. An unstable situation was always created if it was insisted that the core mass contain a larger amount of matter than given by these solutions. For an initial adiabat characterized by a temperature of 450°K and a pressure of 5 × 10^?6 atm, the maximum core mass at which instability occurs is approximately 115 earth masses; this value is rather insensitive to the position in the solar nebula or to the background pressure of the solar nebula. However, if the adiabat is lowered, then the core mass corresponding to instability is decreased. Since the core masses found by Podolak and Cameron for the giant planets are significantly less than the critical core mass corresponding to the initial solar nebula adiabat, we conclude that the giant planets obtained their large amounts of hydrogen and helium by a hydrodynamic collapse process in the solar nebula only after the nebula had been subjected to a considerable period of cooling.     

Accumulation processes in the primitive solar nebula

Particle accumulation processes are discussed for a variety of physical environments, ranging from the collapse phase of an interstellar cloud to the different parts of the models of the primitive solar nebula constructed by Cameron and Pine. Because of turbulence in the collapsing interstellar gas, it is concluded that interstellar grains accumulate into bodies with radii of a few tens of centimeters before the outer parts of the solar nebula are formed. These bodies can descend quite rapidly through the gas toward midplane of the nebula, and accumulation to planetary size can occur in a few thousand years. Substantial modifications of these processes take place in the outer convection zone of the solar nebula, but again it is concluded that bodies in that zone can grow to planetary size in a few thousand years.From the discussion of the interstellar collapse phase it is concluded that the angular momentum of the primitive solar nebula was predominantly of random turbulent origin, and that it is plausible that the primitive solar nebula should have possessed satellite nebulae in highly elliptical orbits. It is proposed that the comets were formed in these satellite nebulae.A number of other detailed conclusions are drawn from the analysis. It is shown to be plausible that an iron-rich planet should be formed in the inner part of the outer nebular convection zone. Discussions are given of the processes of planetary gas accretion, the formation of satellites, the T Tauri solar wind, and the dissipation of excess condensed material after the nebular gases have been removed by the T Tauri solar wind. It is shown that the present radial distances of the planets (but not Bode's Law) should be predicted reasonably well by a solar nebula model intermediate between the uniform and linear cases of Cameron and Pine.     

系外类地行星的内部结构及大气成分研究进展

系外类地行星是目前搜寻地外生命的主要目标.随着观测仪器的发展,现在已经能探测到低于10个地球质量的系外行星.该文简要回顾了系外类地行星的形成与演化,介绍了当前研究它们内部结构的模型和方法,以及由此得出的类地行星质量-半径关系.同时,对应不同的行星初始物质成分,讨论了各种可能的大气结构.最后介绍了未来的空间任务在相关方面的工作.     

Origin of the atmospheres of the terrestrial planets

The monotonic decrease in the atmospheric abundance of ³⁶Ar per gram of planet in the sequence, Venus, Earth, and Mars has been assumed to reflect some conditions in the primitive solar nebula at the time of formation of the planetary atmospheres, having to do either with the composition of the nebula itself or the composition of the trapped gases in small solid bodies in the nebula. Behind such hypotheses lies the assumption that planetary atmospheres steadily gain components. However, not only can gases enter atmospheres; they may also be lost from atmospheres both by adsorption into the planetary interior and by loss into space as a result of collisions with minor and major planetesimals. In this paper a necessarily qualitative discussion is given of the problem of collisions with minor planetesimals, a process called atmospheric cratering or atmospheric erosion, and a discussion is given of atmospheric loss accompanying collision of a planet with a major planetesimal, such as may have produced the Earth's Moon.     

Outline of a theory of planet formation in binary systems

A theory is presented for determining regions where planets may form in binary star systems. Planet formation by accretion is assumed possible if mean planetesimal collision velocities do not exceed a critical value. Collision velocities are increased by perturbations due to the companion star, treated by secular perturbation theory. Collision velocities are damped by aerodynamic drag within the solar nebula, taken as the linear case of Cameron and Pine.A general feature of planetary systems in binary stars is the existence of two zones. The inner zone has enough damping to permit unimpeded growth by accretion; in the outer zone, growth proceeds to a limited diameter, beyond which damping is insufficient. It is shown that the asteroids could not have failed to coalesce due to Jupiter perturbations in the primitive solar nebula. Binary star systems with semimajor axis < 30AU are not likely to have planets; these include Alpha Centauri and 70 Ophiuchi. Systems possibly possessing planets include Eta Cassiopeiae, 40 Eridani, and Σ 2398. Epsilon Eridani is a marginal case.     

Formation of an iron-poor Moon by partial capture,or: Yet another exotic theory of lunar origin

It is shown how it is possible to explain the low abundance of iron and siderophiles in the Moon in a natural way. This is done by an extension of Öpik's mechanism, whereby one or more planetoids pass through Earth's Roche zone, are broken up, and have part of their material captured. Assuming the planetoids are differentiated, the iron core can easily escape capture. This process does not involve any dissipation mechanisms and goes a long way toward explaining the peculiar bulk composition of the Moon. The picture is consistent with that of Urey, in which the early solar system contained more (and smaller) planets than it does today. In the second part of the paper, dynamical considerations are applied for two models, and it is shown that capture from one (or a few) planetoids is quite feasible, whereas a large number of contributors is highly unlikely.     

The moon as a high temperature condensate

The accretion during condensation mechanism, if it occurs during the early over-luminous stage of the Sun, can explain the differences in composition of the terrestrial planets and the Moon. An important factor is the variation of pressure and temperature with distance from the Sun, and in the case of the Moon and captured satellites of other planets, with distance from the median plane. Current estimates of the temperature and pressure in the solar nebula suggest that condensation will not be complete in the vicinity of the terrestrial planets, and that depending on location, iron, magnesium silicates and the volatiles will be at least partially held in the gaseous phase and subject to separation from the dust by solar wind and magnetic effects associated with the transfer of angular momentum just before the Sun joins the Main Sequence.Many of the properties of the Moon, including the enrichment in Ca, Al, Ti, U, Th, Ba, Sr and the REE and the depletion in Fe, Rb, K, Na and other volatiles can be understood if the Moon represents a high temperature condensate from the solar nebula. Thermodynamic calculations show that Ca, Al and Ti rich compounds condense first in a cooling nebula. The high temperature mineralogy is gehlenite, spinel, perovskite, Ca-Al-rich pyroxenes and anorthite. The model is consistent with extensive early melting, shallow melting at 3 AE and with presently high deep internal temperatures. It is predicted that the outer 250 km is rich in plagioclase and FeO. The low iron content of the interior in this model raises the interior temperatures estimated from electrical conductivity by some 800°C. The lunar crust is 80% gabbroic anorthosite, 20% basalt and is about 250-270 km thick. The lunar mantle is probably composed of spinel, merwinite and diopside with a density of 3.4 g cm^–3.Paper dedicated to Prof. Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.Contribution No. 2260, Division of Geological and Planetary Sciences California Institute of Technology, Pasadena, Calif. 91109, U.S.A. Presented at theIAU Symp. Cosmochem., Cambridge, Mass. August 14-16, 1972.     

Migration of giant planets in a time-dependent planetesimal accretion disc

In this paper we develop further the model for the migration of planets introduced in Del Popolo et al. We first model the protoplanetary nebula as a time-dependent accretion disc, and find self-similar solutions to the equations of the accretion disc that give us explicit formulae for the spatial structure and the temporal evolution of the nebula. These equations are then used to obtain the migration rate of the planet in the planetesimal disc, and to study how the migration rate depends on the disc mass, on its time evolution and on some values of the dimensionless viscosity parameter α . We find that planets that are embedded in planetesimal discs, having total mass of  10^-4-0.1 M_⊙  , can migrate inward a large distance for low values of α (e.g.,   α ≃10^-3-10^-2)  and/or large disc mass, and can survive only if the inner disc is truncated or because of tidal interaction with the star. Orbits with larger a are obtained for smaller values of the disc mass and/or for larger values of α . This model may explain several orbital features of the recently discovered giant planets orbiting nearby stars.     

Gas capture and rare gas retention by accreting planets in the solar nebula

In this paper, the physico-chemical effects of the nebula gas on the planets are reviewed from a standpoint of planetary formation in the solar nebula.The proto-Earth growing in the nebula was surrounded by a primordial atmosphere with a solar chemical composition and solar isotopic composition. When the mass of the proto-Earth was greater than 0.3 times the present Earth mass, the surface was molten because of the blanketing effect of the atmosphere. Therefore, the primordial rare gasses contained in the primordial atmosphere dissolved into the molten Earth material without fractionation and in particular the dissolved neon is expected to be conserved in the present Earth material. Hence, if dissolved neon with a solar isotopic ratio is discovered in the Earth material, it will indicate that the Earth was formed in the nebula and that the dissolved rare gases were one of the sources which degassed to form the present atmosphere.     

Earth and Venus: A comparative study

The observed density of Venus is about 2% smaller than would be expected if Venus were a twin planet of the Earth, possessing an identical internal composition and structure. In principle, this could be explained by a process of physical segregation of metal particles from silicate particles in the solar nebula prior to accretion, so that Venus accreted from relatively metal-depleted material. However, this model encounters severe difficulties in explaining the nature of the physical segregation process and also the detailed chemical composition of the Earth's mantle. Two alternative hypotheses are examined, both of which attempt to explain the density difference in terms of chemical fractionation processes. Both of these hypotheses assume that the relative abundances of the major elements Fe, Si, Mg, Al, and Ca are similar in both planets. According to the first hypothesis, a larger proportion of the total iron in Venus is present as iron oxide in the mantle, so that the core-to-mantle ratio is smaller than in the Earth. This model implies that Venus is more oxidized than the Earth, with its lower intrinsic density (i.e., corrected to equivalent pressures and temperatures) due to the larger amount of oxygen present. The difference between oxidation states is attributed to differing degrees of accretional heating arising from the relatively smaller mass of Venus. On the other hand, the second hypothesis maintains that Venus is more reduced than the Earth, with its mantle essentially devoid of oxidized iron. The difference intrinsic densities is attributed to the Earth accreting at a lower temperature than Venus as a result of the Earth's greater distance from the center of the nebula. As a result, large amounts of sulfur accreted on the Earth but not on Venus. The sulfur, which entered the core, is believed to have increased the mean density of the Earth because of its relatively high atomic weight. The hypothesis also implies that most of the Earth's potassium, because of its chalcophile properties, entered the core.These hypotheses are evaluated in the light of existing data. The second hypothesis leads to an intrinsic density for Venus which is only 0.4% smaller than that of the Earth. This difference is much smaller than is believed to exist. A wide range of chemical evidence is found to be unfavorable to this second hypothesis, but to be consistent with the interpretation that Venus is more oxidized than the Earth, as required by the first hypothesis.     

Temperature distribution in the solar nebula at successive stages of its evolution

We have constructed a model of the solar nebula that allows for the temperature and pressure distributions at various stages of its evolution to be calculated. The mass flux from the accretion envelope to the disk and from the disk to the Sun, the turbulent viscosity parameter α, the opacity of the disk material, and the initial angular momentum of the protosun are the input model parameters that are varied. We also take into account the changes in the luminosity and radius of the young Sun. The input model parameters are based mostly on data obtained from observations of young solar-type stars with disks. To correct the input parameters, we use the mass and chemical composition of Jupiter, as well as models of its internal structure and formation that allow constraints to be imposed on the temperature and surface density of the protoplanetary disk in Jupiter’s formation zone. Given the derived constraints on the input parameters, we have calculated models of the solar nebula at successive stages of its evolution: the formation inside the accretion envelope, the evolution around the young Sun going through the T Tauri stage, and the formation and compaction of a thin dust layer (subdisk) in the disk midplane. We have found the following evolutionary trend: an increase in the temperature of the disk at the stage of its formation, cooling at the T Tauri stage, and the subsequent internal heating of the dust subdisk by turbulence dissipation that causes a temperature rise in the formation zone of the terrestrial planets at the high subdisk density and the opacity in this zone. We have obtained the probable ranges of temperatures in the disk midplane, i.e., the temperatures of the protoplanetary material in the formation region of the terrestrial planets at the initial stage of their formation.     

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.     

Extraterrestrial microspherules and iron needles in the interstellar medium

Microspherules in Paleozoic-Mesozoic bedded chert, presumed to be of extraterrestrial origin and comprised of clusters of iron needles,may represent an early stage in the aggregation of the inner planets. A fraction of the constituent iron needles will be expected to be carried along with expanding gas to the periphery of the solar nebula where the outer planets and comets condensed. A substantial fraction of iron needles would escape into interstellar space and add to the population of interstellar grains. Iron needles could contribute substantially to interstellar polarization and to the infrared opacity of the interstellar medium.     

Orbital resonances in the solar nebula: Implications for planetary accretion

The influence of gas drag and gravitational perturbations by a planetary embryo on the orbit of a planetesimal in the solar nebula was examined. Non-Keplerian rotation of the gas causes secular decay of the orbit. If the planetesimal's orbit is exterior to the perturber's, resonant perturbations oppose this drag and can cause it to be trapped in a stable orbit at a commensurability of order j/(j + 1), where j is an integer. Numerical and analytical demonstrations show that resonant trapping occurs for wide ranges of perturbing mass, planetesimal size, and j. Induced eccentricities are large, causing overlap of orbits for bodies in different resonances with j > 2. Collisions between planetesimals in different resonances, or between resonant and nonresonant bodies, result in their disruption. Fragments smaller than a critical size can pass through resonances under the influence of drag and be accreted by the embryo. This effect speeds accretion and tends to prevent dynamical isolation of planetary embryos, making gas-rich scenarios for planetary formation more plausible.     

The formation of iron,stone, and mixed planetesimals in the early solar system

The interaction of dust grains with each other in a finite-temperature solar nebula are examined, taking into account the important fact that such grains would carry net steady-state charges like those of grains in interstellar clouds. This charge is given by the well-known Spitzer relation. It provides a screening mechanism that operates during accretion and results in bodies of differing compositions depending on the local temperature in the nebula. In a typical nebula, it is found that planetesimals of 0.1–10²-cm size form in a time of order 10⁶–10⁷ years. These planetesimals are of iron and stone and mixed composition in the inner solar system, but of mixed composition only in the outer solar system. The predictions of this type of charged-dust accretion can be compared to known data on meteorites and the composition of the planets.