Radial migration of planetesimals

Planetesimals orbiting a protostar in a circumstellar disk are affected by gravitational interaction among themselves and by gas drag force due to disk gas. Within the Kyoto model of planetesimal accretion, the migration rate is interpreted as the inverse of the planetary formation time scale. Here, we study time scales of gravitational interaction and gas drag force and their influence on planetesimal migration in detail. Evaluating observations of 86 T Tauri stars (Beckwithet al., 1990), we find the mean radial temperature profile of circumstellar disks. The disk mass is taken to be 0.01M in accordance with minimum mass models and observed T Tauri disks. The time scale of gravitational interaction between planetesimals is studied analogously to Chandrasekhar's stellar dynamics. Hence, Chandrasekhar's coefficient , defined as the fraction between the mean separation of planetesimals and the impact parameter, plays an important role in determining the migration rate. We find ln to lie between 5 and 10 within the protosolar disk. Our result is that, at the stage of disk evolution considered here, gas drag force affects the radial migration of planetesimals by a few orders of magnitude more than gravitational interaction.Paper presented at the Conference on Planetary Systems: Formation, Evolution, and Detection held 7–10 December, 1992 at CalTech, Pasadena, California, U.S.A.     

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.     

Collisional growth of planetesimals

Safronov's (1972) demonstration that relative velocities of planetesimals would be comparable to the dominant size bodies' escape velocities, combined with a plausible size distribution that has most mass in the largest bodies, yielded his evolution model with limited growth of the largest planetesimal with respect to its next largest neighbors. A numerical simulation of planetesimal accretion (Greenberget al., 1978) suggests that at least over one stage of collisional accretion, velocities were much lower than the escape velocity of the largest bodies, because the bulk of the mass still resided in km-scale bodies. The low velocities at this early stage may conceivably have permitted early runaway growth, which, in turn, would have kept the velocities low and permitted continued runaway growth of the largest bodies.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

Initial sizes of planetesimals and accretion of the asteroids

The present size frequency distribution (SFD) of bodies in the asteroid belt appears to have preserved some record of the primordial population, with an excess of bodies of diameter D ∼ 100 km relative to a simple power law. The survival of Vesta’s basaltic crust also implies that the early SFD had a shallow slope in the range ∼10-100 km. (Morbidelli, A., Bottke, W.F., Nesvorny, D., Levison, H.F. [2009]. Icarus 204, 558-573) were unable to produce these features by accretion from an initial population of km-sized planetesimals. They concluded that bodies with sizes in the range ∼100-1000 km and a SFD similar to the current population were produced directly from solid particles of sub-meter scale, without experiencing accretion through intermediate sizes. We present results of new accretion simulations in the primordial asteroid region. The requisite SFD can be produced from an initial population of planetesimals of sizes ?0.1 km, smaller than the usual assumption of km-sized bodies. The bump at D ∼ 100 km is produced by a transition from dispersion-dominated runaway growth to a regime dominated by Keplerian shear, before the formation of large protoplanetary embryos. Thus, accretion of the asteroids from an initial population of small (sub-km) planetesimals cannot be ruled out.     

Potassium diffusion in melilite: Experimental studies and constraints on the thermal history and size of planetesimals hosting CAIs

Abstract— Among the calcium‐aluminum‐rich inclusions (CAIs), excess ⁴¹K (⁴¹K*), which was produced by the decay of the short‐lived radionuclide ⁴¹Ca (t_1/2 = 0.1 Myr), has so far been detected in fassaite and in two grains of melilites. These observations could be used to provide important constraints on the thermal history and size of the planetesimals into which the CAIs were incorporated, provided the diffusion kinetic properties of K in these minerals are known. Thus, we have experimentally determined K diffusion kinetics in the melilite end‐members, åkermanite and gehlenite, as a function of temperature (900–1200 °C) and crystallographic orientation at 1 bar pressure. The closure temperature of K diffusion in melilite, T_c(K:mel), for the observed grain size of melilite in the CAIs and cooling rate of 10–100 °C/Myr, as calculated from our diffusion data, is much higher than that of Mg in anorthite. The latter was calculated from the available Mg diffusion data in anorthite. Assuming that the planetesimals were heated by the decay of ²⁶Al and ⁶⁰Fe, we have calculated the size of a planetesimal as a function of the accretion time t_f such that the peak temperature at a specified radial distance r_c equals T_c(K:mel). The ratio (r_c/R)³ defines the planetesimal volume fraction within which ⁴¹K* in melilite grains would be at least partly disturbed, if these were randomly distributed within a planetesimal. A similar calculation was also carried out to define R versus t_f relation such that ²⁶Mg* was lost from ?50% of randomly distributed anorthite grains, as seems to be suggested by the observational data. These calculations suggest that ?60% of melilite grains should retain ⁴¹K* if ?50% of anorthite grains had retained ²⁶Mg*. Assuming that t_f was not smaller than the time of chondrule formation, our calculations yield minimum planetesimal radius of ?20–30 km, depending on the choice of planetesimal surface temperature and initial abundance of the heat producing isotope ⁶⁰Fe.     

Satellite-sized planetesimals and lunar origin

Exploratory calculations using accretionary theory are made to demonstrate plausible sizes of second-largest, third-largest, etc., bodies at the close of planet formation in heliocentric orbits near the planets, assuming asteroid-like size distributions at the start of the calculation. Many satellite-sized bodies are found to be available for capture, cratering, or collisional fragmentation. In the case of Earth-sized planets, the models suggest second-largest bodies of 500 to 3000 km radius, and tens of bodies larger than 100 km radius. Many of these interact with the planet before suffering any fragmentation events with each other. Collision of a large body with Earth could eject iron-deficient crust and upper mantle material, forming a cloud of refractory, volatile-poor dust that could form the Moon. Other satellite systems may have been affected by major capture or collision events of chance character.     

Formation and properties of fluffy planetesimals

A mechanism of accumulation of grains in the primordial solar nebula is described. This process produces porous, low density compressible aggregates. Compaction of the aggregates in a collision between them dissipates the kinetic energy of the collision and can result in efficient growth. A simple analysis of such collisions is developed and applied over a range of aggregate sizes and relative velocities. The results indicate that large planetesimals could grow through collisions rather than fragment if the conditions are favorable. Our modelling suggests that primordial asteroids and comets on the order of a kilometer in size will have low densities and irregular shapes.Paper presented at the Conference on Planetary Systems: Formation, Evolution, and Detection held 7–10 December, 1992 at CalTech, Pasadena, California, U.S.A.     

Capture of planetesimals by the giant planets

We investigate the possibility of gravitational capture of planetesimals as temporary or permanent satellites of Uranus and Neptune during the process of planetary growth. The capture mechanism is based in the enhancement of the Hill's sphere of action not only due to the mass acquired by the planet, but also by the variation of the planet-Sun distance as a consequence of the scattering of planetesimals by the planets of the outer solar system. Our calculations indicate that satellite capture was very important, specially during the first stages of the accretion process, contributing in a significant way to the planetary growth.     

Accumulation of planetesimals in the solar nebula

Characteristic time scales relevant to the accumulation of planetesimals in a gaseous nebula are examined and the accumulation toward the planets is simulated by numerically solving a growth equation for a mass distribution function. The eccentricity and inclination of planetesimals are assumed to be determined by a balance between excitation due to mutual gravitational scattering and dissipation due to gas drag. Two kinds of mass motion in the radial direction, i.e., diffusion due to mutual scattering and inward flow due to gas drag, are both taken into account. The diffusion is shown to be effective in later stages with a result of accelerating the accumulation. As to the coalescent collision cross section, the usual formula for a binary encounter in a free space is used but the effect of tidal disruption which increases substantially the cross section is taken into account. Numerical results show that the gravitational enhancement factor (i.e., the so-called “Safronov number”), contained in the cross section formula, always takes a value of the order of unity but the accumulation proceeds relatively rapidly owing to the effects of radial diffusion and tidal disruption. That is, a proto-Earth, a proto-Jupiter, and a proto-Saturn with masses of 1×10²⁷ g are formed in 5×10⁶, 1×10⁷, and 1.6×10⁸ years, respectively. Also, a tentative numerical computation for the Neptune formation shows that a proto-Neptune with the same mass requires a long accumulation time, 4.6×10⁹ years. Finally, the other effects which are expected to reduce the above growth times further are discussed.     

Tidal disruption of dissipative planetesimals

Tidal disruption is a potentially important process for the accumulation of the planets from planetesimals. The fact that stable equilibria do not exist for circular orbits inside the Roche limit has often been hypothesized to mean that any object that passes within the Roche limit is totally disrupted. We have disproven this hypothesis by solving the dynamic problem of the tidal disruption of a dissipative planetestimal during a close encounter with a protoplanet. The solution consists of a numerical integration of the three-dimensional, nonlinear equations of motion, including an approximate treatment of viscous dissipation in the solid regions of the planetesimal. The numerical methods have been extensively tested on a series of one-, two- (Jeans), and three-(Roche) dimensional test problems involving the equilibrium of a body subjected to tidal forces. The results may be scaled to planetesimals of arbitrary size, providing that the scaled equation of state applied. The calculations show that a strongly dissipative planetesimal which passes by the Earth on a parabolic orbit with a perigee within the Roche limit (≈3R_Earth) is not tidally disrupted (even for grazing incidence), and loses no more than a few percent of its mass. This result applies to bodies of radius R which have a kinematic viscosity ν ? 10¹²(R/1000km)² cm²sec^?1. Less dissipative planetesimals (ν ≈ 10¹³(R/1000 km)² cm²sec^?1) may lose up to about 20% of their mass. There are two coupled reasons why this result differs from previous hypotheses: (1) in a dynamic encounter, there is insufficient time to disrupt the planetesimal, and (2) even in circular orbit, the small velocities in the solid region imply that many orbital periods are necessary to completely disrupt the planetesimal. Hence solid and partially molten planetesimals will not experience substantial tidal disruption; completely molten bodies may be sufficiently inviscid to undergo tidal disruption.     

Some consequences of a steady thermal moon

The Moon is represented as an inhomogeneous spherical body in a steady thermal state. Radioactive heat sources are supposed distributed in a manner which is consistent both with the total measured heat flux near the surface and with the broad seismic evidence. Surface concentrations of uranium and thorium are those suggested by the study of Apollo 11 samples. The resultant internal temperature profile allows the details of Sonett's electrical conductivity profile to be understood if it is accepted that the Moon was not cold 4.5 × 10⁹ yr ago. It would appear further that at least one of the maria was formed by the impact of planetesimals.     

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.     

Chondrule formation by the ablation of small planetesimals

Abstract— Numerous models have been proposed to explain the formation of chondrules, but none can be reconciled with the highly diverse properties of these objects. Here the formation of chondrules by the surface melting and ablation of small planetesimals in nebula shock waves is investigated using a numerical model. It is shown that bodies between ～1 mm and 500 m in diameter would have produced molten droplets by ablation during gas drag in nebula shocks stronger than ～2.0 Mach. The properties of chondrules produced by ablation are estimated by comparison with meteorite fusion crusts and through consideration of the environment within the bow shock envelope of ablating planetesimals. It is suggested that most ablation chondrules will have broadly chondritic compositions with depletions in siderophile and chalcophile elements and relatively high volatile contents and textures that are mainly porphyritic. The formation of chondrules by ablation of planetesimals in shock waves was probably most important at a late stage in nebula history and occurred at the same time as chondrules formed by the melting of dust particles. The high abundance of dust particles relative to larger bodies at all stages of accretion implies that only a proportion of chondrules may have been formed by ablation and that genetic groups of chondrules with very different origins may coexist in meteorites.     

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.     

The rotation rates of accreting planetesimals

There are obtained upper limits for the relative velocity at infinity of accreting planetesimals for a nearly constant mass of the largest accreting planetesimal and also in the case of variable mass. We conclude, that while the larger planets cannot be brought to the stage of rotational instability by stochastic collisions, the asteroids could be brought. provided that the relative velocities in the asteroid belt were larger than about 2 km s^–1.     

Collisions in self-gravitating clouds of planetesimals

A theory of partially elastic collisions is constructed for frictionless planetesimals in an arbitrary gravitational field. The non-zero size of the particles and the influence of gravitational encounters are included. The equations for a self-gravitating rotationally symmetric disk or ring are written in an explicit form. Such systems turn out to be bimodal in the same sense as the Keplerian systems, i.e. there are two kinds of stable configurations which may co-exist in adjacent regions without disturbing the mechanical equilibrium. The transitions from one mode to another can also occur at essentially smaller values of the optical thickness than those previously found for Saturn's rings: in one of the numerically studied cases the transition from the dense to the rarefied mode occurred at the optical thickness 3×10^?5 while the reversed process corresponded to a higher value, 10^?2. The difference illustrates the dependence of the transition on its direction. The characteristic S shape which several authors have found for the relation between the viscosity and the optical thickness in Keplerian systems becomes more complicated if the contribution of self-gravitation increases. In some cases the stable solutions also imply a certain minimum value of the optical thickness.     

Growth of planetesimals and clearance of disks around the T-tauri stars

A growth and contraction of dust condensations formed as a result of development of gravitation instability in a dust subdisk is discussed within the framework of the generally accepted scheme of evolution of a circumsolar pre-planetary disk. The time of evolution of condensations necessary for clearing the hypothetical disks around young stars of the Sun type is estimated.     

From icy planetesimals to outer planets and comets

Numerical simulations of planet growth in the outer solar system shows thatgrwoth of Uranus and Neptune occurs in reasonably short time, well below the actual age of the system, without the need for ad hoc assumptions about excess mass or artificially low relative velocities among the icy planetesimals. Low velocities, which speed accretion, are a natural consequence of the non-power-law size distribution of planetesimals, just as in our earlier simulations of terrestial planet growth. Initial planetesimals of size ～ 100 km, predicted by formal expressions for gravitational instability in a thin disk of solid material, failed to produce sufficient debris in the size range 1 to 10 km to account for population of the Oort cloud with comet-sized bodies. However, our model of nonhomologous settling of grains to the midplane of the solar system shows that gravitational clumping did not wait until all solid material had settled to the midplane, as had been assumed in earlier models. Rather, the clumping occurred in successive portions of the material that reached the midplane, producing “initial” planetesimals probably of comet-like sizes. Models of subsequent collisional evolution show that such an initial size distribution, similar to known comets, would have been required in order to have an adequate comet-like size distribution available to feed the Oort cloud as the other planets reach full size. Comets are probably unaltered remnants of the initial population of planetesimals in the outer solar system, not fragments of larger bodies.