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.     

Stochastic coalescence model for terrestrial planetary accretion

A nonequilibrium stochastic coalescence model for terrestrial planetary accretion is developed by using an approximation to the Safronov-Golovin solution for the scalar transport equation with linear kernel. According to this model, formation of comparatively massive objects occurs quite rapidly during the early stages of accretive evolution in a given terrestrial planetesimal population, while during late growth stages, an increasingly substantial fraction of total population mass becomes incorporated into progressively fewer, relatively very large bodies. The model also implies that the (conservative) growth rate of the population's largest member varies directly as its mass, and further suggests that this growth rate may not decline significantly until very nearly final planetary mass is attained.     

Star formation sustained by gas accretion

Numerical simulations predict that metal-poor gas accretion from the cosmic web fuels the formation of disk galaxies. This paper discusses how cosmic gas accretion controls star formation, and summarizes the physical properties expected for the cosmic gas accreted by galaxies. The paper also collects observational evidence for gas accretion sustaining star formation. It reviews evidence inferred from neutral and ionized hydrogen, as well as from stars. A number of properties characterizing large samples of star-forming galaxies can be explained by metal-poor gas accretion, in particular, the relationship among stellar mass, metallicity, and star-formation rate (the so-called fundamental metallicity relationship). They are put forward and analyzed. Theory predicts gas accretion to be particularly important at high redshift, so indications based on distant objects are reviewed, including the global star-formation history of the universe, and the gas around galaxies as inferred from absorption features in the spectra of background sources.     

Circumstellar disks and planetary formation

The role of circumstellar disks in star and planetary formation is briefly reviewed. The observed disk around MWC 349 is used as an example and a table of evolutionary time scales and parameters is presented. The disk about MWC 349 is characteristic of that expected about a massive star. Disk structure about solar mass stars is more completely reviewed by Cameron (1978). The parameters for the disk indicate that there is a deduced region where conditions are appropriate for dust condensation and possible aggregation of material to planetary masses. For the purposes of the discussion we are assuming that the infrared as well as optical radiation arises from the disk which extends the known extent of the disk to 10¹⁴cm. It is not yet certain that this is the case.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.     

Numerical modeling of protocore destabilization during planetary accretion: Methodology and results

We developed and tested an efficient 2D numerical methodology for modeling gravitational redistribution processes in a quasi spherical planetary body based on a simple Cartesian grid. This methodology allows one to implement large viscosity contrasts and to handle properly a free surface and self-gravitation. With this novel method we investigated in a simplified way the evolution of gravitationally unstable global three-layer structures in the interiors of large metal-silicate planetary bodies like those suggested by previous models of cold accretion [Sasaki, S., Nakazawa, K., 1986. J. Geophys. Res. 91, 9231-9238; Karato, S., Murthy, V.R., 1997. Phys. Earth Planet Interios 100, 61-79; Senshu, H., Kuramoto, K., Matsui, T., 2002. J. Geophys. Res. 107 (E12), 5118. 10.1029/2001JE001819]: an innermost solid protocore (either undifferentiated or partly differentiated), an intermediate metal-rich layer (either continuous or disrupted), and an outermost silicate-rich layer. Long-wavelength (degree-one) instability of this three-layer structure may strongly contribute to core formation dynamics by triggering planetary-scale gravitational redistribution processes. We studied possible geometrical modes of the resulting planetary reshaping using scaled 2D numerical experiments for self-gravitating planetary bodies with Mercury-, Mars- and Earth-size. In our simplified model the viscosity of each material remains constant during the experiment and rheological effects of gravitational energy dissipation are not taken into account. However, in contrast to a previously conducted numerical study [Honda, R., Mizutani, H., Yamamoto, T., 1993. J. Geophys. Res. 98, 2075-2089] we explored a freely deformable planetary surface and a broad range of viscosity ratios between the metallic layer and the protocore (0.001-1000) as well as between the silicate layer and the protocore (0.001-1000). An important new prediction from our study is that realistic modes of planetary reshaping characterized by a high viscosity protocore and low viscosity molten silicate and metal [Senshu, H., Kuramoto, K., Matsui, T., 2002. J. Geophys. Res. 107 (E12), 5118. 10.1029/2001JE001819] may result in the transient exposure of the protocore to the planetary surface and a strongly (up to 8% of the planetary diameter) aspherical deviation of the planetary shape during the early stages of core formation. Exposure of the protocore might happen in the early stages of iron core formation. This process may conceivably convert a large amount of potential energy into temperature increase and a transient strongly non-uniform depth of the magma ocean around the protoplanet. Our simplified model also predicts that the time for metallic core formation out of the metal-rich layer depends mainly on the dynamics of the deformation of the solid strong protocore. In nature this dynamics will be strongly dependent on the effective viscosity of the protocore, which should generally have non-Newtonian pressure-, temperature-, and stress-dependent rheology with strong thermomechanical feedbacks from gravitational energy dissipation.     

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.     

Orbital resonances and planetary formation sites

Orbital resonances may have played an important role in determining the locations where the planetesimal swarm eventually accreted into full-size planets. Several pairs of planets do indeed have commensurable orbital periods at present, but the case for control of planet formation by resonances is weakened by the fact that many pairs are not commensurable and that those which are do not necessarily exist at the strongest resonances. However, the mass loss and redistribution that occurred in the early solar system evolution can substantially alter the positions of planets and planetary embryos within the swarm. A cascaded resonance structure is hypothesized where planetesimal growth was accelerated at 2:1 interior and 1:2 exterior resonances with an early-formed Jupiter producing runaway growth of planetary embryos. These embryos produce their own resonances which, in turn, lead to additional embryos in a process that successively propagates inward and outward to generate a resonant configuration of embryos. In this manner, the early presence of Jupiter imposed a harmonic structure on the accumulating planetesimal swarm. For an accretion disk with surface density obeying a power law of index ?1.2 the positions of the planetary embryos can be moved into a reasonably good agreement with most of the present planetary positions that is as good as that given by the Titius-Bode law.     

Metallicity, planetary formation and migration

Recent observations show a clear correlation between the probability of hosting a planet and the metallicity of the parent star. As radial velocity surveys are biased, however, towards detecting planets with short orbital periods, the probability–metallicity correlation could merely reflect a dependence of migration rates on metallicity. We investigated the possibility, but find no basis to suggest that the migration process is sensitive to the metallicity. The indication is, therefore, that a higher metallicity results in a higher probability for planet  formation .     

Computer simulations of planetary accretion dynamics: Sensitivity to initial conditions

We have tested the implications and limitations of Program ACRETE, a scheme based merely on Newtonian physics and accretion with unit sticking efficiency, devised by Dole in 1970 to simulate the origin of the planets. The dependence of the results on a variety of radial and vertical density distribution laws, on the ratio of gas to dust in the solar nebula, on the total nebula mass, and on the orbital eccentricity, ?, of the accreting grains are explored. Only for a small subset of conceivable cases are planetary systems closely like our own generated. Many models have tendencies toward one of two preferred configurations: multiple-star systems, or planetary systems in which Jovian planets either have substantially smaller masses than in our system or are absent altogether. But for a wide range of cases recognizable planetary systems are generated, ranging from multiple-star systems with accompanying planets, to systems with Jovian planets at several hundred astronomical units, to single stars surrounded only by asteroids. Many systems exhibit planets like Pluto and objects of asteroidal mass, in addition to usual terrestrial and Jovian planets. No terrestrial planets were generated more massive than five Earth masses. The number of planets per system is for most cases of order 10, and, roughly, inversely proportional to ?. All systems generated obey a relation of the Titius-Bode variety for relative planetary spacing. The case with which planetary systems are generated using such elementary and incomplete physical assumptions supports the idea of abundant and morphologically diverse planetary systems throughout the Galaxy.     

Protocore destabilization in planetary embryos formed by cold accretion: Feedbacks from non-Newtonian rheology and energy dissipation

It is suggested that a distinct population of Mars-sized planetary embryos can form by cold accretion in the outer part of the terrestrial zone. A possible radial structure for such embryos consists of an outer silicate-rich layer and an undifferentiated solid protocore. The protocore is divided in a cold undifferentiated central region, a layer of iron, and an outer shell of undifferentiated material. This structure is gravitationally unstable. We investigate the influence of the rheology of the silicate-rich material and of the thermal effects on the protocore destabilization and on the iron core formation. We use a 2D thermomechanical numerical model of a self-gravitating planetary body. The planetary surface is treated as a free surface by imposing a massless weak medium. Our calculations include a non-Newtonian temperature-, pressure-, and strain rate-dependent viscoplastic rheology and thermal contributions from shear heating. We explored the influence of rheology by varying the activation volume, friction angle, and the maximum yield strength of silicate-rich layers. We distinguished three different core formation regimes: exposure mode, fragmentation mode, and transition mode. Like in previous models with Newtonian rheology, the core experiences large deviations from the spherical shape and is temporarily exposed at the surface in the exposure mode. In contrast to the Newtonian models the destruction of the protocores in the fragmentation modes is driven by (i) the spontaneous strain localization along planetary-scale shear zones forming inside the protocore, and/or (ii) descending localized iron diapirs/sheets penetrating the protocore. Feedback from energy dissipation notably modifies the thermal structure of the deforming planetary body. In particular it causes a temperature increase of up to several hundred Kelvin (i) around the moving and deforming protocore and (ii) along planetary scale rupture zones that form inside the protocore. If the protocore is large and has a high viscosity, a large fraction of the dissipated heat is used to increase the temperature of iron.     

The solar system,planetary systems of stars,and sequential accretion theory

A comparison of solar system properties to experimental data derived from observations of exoplanets results in a significant revision of the accretion theory. We describe some new ideas of the present-day theory of the origin of planets and planetary systems.     

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.     

The floccule theory for planetary formation

The accumulation of floccules into protoplanets is discussed, and it is pointed out that the simplifications which have been introduced into recent numerical models may result in the incorrect conclusion being reached.