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.     

Settling and growth of dust particles in a laminar phase of a low-mass solar nebula

It is considered that some vertical convection as well as possible turbulence in an early phase of solar nebula soon terminates owing to diminution of the temperature dependence of dust opacity due to rapid growth of dust particles. We reexamine settling and growth of dust particles in the subsequent laminar phase of the solar nebula in detail, treating a dust layer as a two-component fluid composed of the dust and the gas. We obtain analytic expressions for the settling path, the growing size, and the settling time. The settling process is divided into two phases, i.e., an early gas-dominant phase and a later dust-dominant phase. So far, only the former phase, where the particle path finally turns from vertical to radial, has been investigated. In the latter phase, dust particles drag the gas, rather than the gas does dust particles. Consequently, the particle path turns from radial to vertical. Dust particles grow most appreciably and rapidly in a radially sweeping phase. The final radii of dust particles at the onset of gravitational instability of the dust layer are 20, 5.9, and 0.60 cm in the Earth's, Jupiter's, and Neptune's zones, respectively. These values are much smaller than those obtained previously by S.J. Weidenschilling [1980, Icarus44, 172–189] and Y. Nakagawa et al. [1981, Icarus45, 517–528]. The total settling times are 1.9 × 10³, 4.6 × 10³, and 2.8 × 10⁴ years in the above-mentioned three zones, respectively. These are somewhat smaller than those obtained by the previous studies. Most of the settling time is spent in the early vertically settling phase.     

Kamacite sulfurization in the solar nebula

Abstract— The kinetics and mechanisms of kamacite sulfurization were studied experimentally at temperatures and H₂S/H₂ ratios relevant to the solar nebula. Pieces of the Canyon Diablo meteorite were heated at 558 K, 613 K, and 643 K in 50 parts per million by volume (ppmv) H₂S-H₂ gas mixtures for up to one month. Optical microscopy and x-ray diffraction analyses show that the morphology and crystal orientation of the resulting sulfide layers vary with both time and temperature. Electron microprobe analyses reveal three distinct phases in the reaction products: monosulfide solid solution (mss), (Fe, Ni, Co)_1-xS, pentlandite (Fe, Ni, Co)_9-xS₈, and a P-rich phase. The bulk composition of the remnant metal was not significantly changed by sulfurization. Kamacite sulfurization at 558 K followed parabolic kinetics for the entire duration of the experiments. Sulfide layers that formed at 613 K grew linearly with time, while those that formed at 643 K initially grew linearly with time then switched to parabolic kinetics upon reaching a critical thickness. The experimental results suggest that a variety of thermodynamic, kinetic, and physical processes control the final composition and morphology of the sulfide layers. We combine morphological, x-ray diffraction, electron microprobe, and kinetic data to produce a comprehensive model of sulfide formation in the solar nebula. Then, we present a set of criteria to assist in the identification of solar nebula condensate sulfides in primitive meteorites.     

Grain motions in the solar nebula

Isotopic analyses of meteorites suggest the possibility that some interaction between supernova ejecta and grains occurred in the solar nebula. In particular, the dynamics of grain motions in the solar nebula can explain the observed mixing of nucleosynthetic components. The effect of a shock wave on the motions of grains are examined. A steady-state, plane shock propagating into a uniform region of gas and dust grains is followed by a zone of gas/grain slip, in which the grains are accelerated by drag forces from the pre-shock to the post-shock gas velocity, i.e. reducing the relative velocity between the gas and grains to zero. On the basis of these calculations, it is estimated that if grains carried the isotopic anomalies investigated by Lee, Papanastassoiu, and Wasserburg (1978), then those grains could be no bigger than 2×10^–4 cm in size. A scenario is suggested in which the sluggishness of grains provides a natural way to concentrate and mix the nucleosynthetic components carried by grains in the ejecta and in the solar nebula.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.     

Contraction of the solar nebula

The concept of Roche limit is applied to the Laplacian theory of the origin of the solar system to study the contraction of a spherical gas cloud (solar nebula). In the process of contraction of the solar nebula, it is assumed that the phenomenon of supersonic turbulent convection described by Prentice (1978) is operative and brings about the halt at various stages of contraction. It is found that the radius of the contracting solar nebula follows Titius-Bode law R _p = Ra^p, where R is the radius of the present Sun and a = 1.442. We call a the Roche's constant. The consequences of the relation are also discussed. The aim, here, is an attempt to explain, on the basis of the concept of Roche limit, the distribution of planets in the solar system and try to understand the physics underlying it.     

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.     

Secular resonances in the primitive solar nebula

We present here a very simple model that could explain the relatively high eccentricities and inclinations observed in the minor planet belt. This model is based upon the sweeping of the secular resonances ₆ and ₁₆ through the belt due to the gravitational effect of the dissipation of a primitive solar nebula. The sweeping of the ₁₆ secular resonance (responsible for the high inclinations) is very sensitive to the density profile of the nebula. For the model to work we need a density profile proportional to ^–k with between 1.0 and 1.5.     

Density waves in the solar nebula: Diffential Lindblad torque

The differential torque between a two-dimensional fluid disc and an embedded perturber due to waves launched at its resonances is calculated. For a non-self-gravitating disc, the disturbance is in the form of pressure waves which propagate away from co-rotation and asymptotically become trailing spirals. The torque calculation includes lowest order contributions from gradients in various disc properties. The orbital drift rate of the perturber is determined and the results applied to the motion of preplanetary material in the solar nebula.     

The solar oxygen-isotopic composition: Predictions and implications for solar nebula processes

Abstract— The outer layers of the Sun are thought to preserve the average isotopic and chemical composition of the solar system. The solar O-isotopic composition is essentially unmeasured, though models based on variations in meteoritic materials yield several predictions. These predictions are reviewed and possible variations on these predictions are explored. In particular, the two-component mixing model of Clayton and Mayeda (1984) (slightly revised here) predicts solar compositions to lie along an extension of the calcium-aluminum-rich inclusion (CAI) ¹⁶O line between (δ¹⁸O, δ¹⁷O) = (16.4, 11.4)%0 and (12.3, 7.5)%0. Consideration of data from ordinary chondrites suggests that the range of predicted solar composition should extend to slightly lower δ¹⁸O values. The predicted solar composition is critically sensitive to the solid/gas ratio in the meteorite-forming region, which is often considered to be significantly enriched over solar composition. A factor of two solid/gas enrichment raises the predicted solar (δ¹⁸O, δ¹⁷O) values along an extension of the CAI ¹⁶O line to (33, 28)%0. The model is also sensitive to the nebular O gas phase. If conversion of most of the gaseous O from CO to H₂O occurred at relatively low temperatures and was incomplete at the time of CM aqueous alteration, the predicted nebular gas composition (and hence the solar composition) would be isotopically heavier along a slope 1/2 line. The likelihood of having a single solid nebular O component is discussed. A distribution of initial solid compositions along the CAI ¹⁶O line (rather than simply as an end-member) would not significantly change the predictions above in at least one scenario. Even considering these variations within the mixing model, the predicted range of solar compositions is distinct from that expected if the meteoritic variations are due to non-mass-dependent fractionation. Thus, a measurement of the solar O composition to a precision of several permil would clearly distinguish between these theories and should clarify a number of other important issues regarding solar system formation.     

Gas flow in the solar nebula leading to the formation of Jupiter

Three-dimensional gas flow in the solar nebula, which is subject to the gravity of the Sun and proto-Jupiter, is numerically calculated by using a three-dimensional hydrodynamic code - i.e., the socalled smoothed-particle method. The flow is circulating around the Sun as well as falling into a potential well of proto-Jupiter. The results for various masses of proto-Jupiter show that (1) the e-folding growth time of proto-Jupiter by accretion of the nebular gas is as short as about 300 years in stages where the mass of proto-Jupiter is 0.2 ~ 0.5 times the present Jovian mass, and that (2) proto-Jupiter begins to push away the nebular gas from the orbit of proto-Jupiter and form a gap around the orbit, when its mass is about 0.7 times the present Jovian mass. It is possible that this pushing-away process determined the present Jovian mass.     

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.     

Numerical models of the primitive solar nebula

Numerical models have been constructed to represent probable conditions in the primitive solar nebula. A two solar mass fragment of a collapsing interstellar gas cloud has been represented by a uniformly rotating sphere. Two cases have been considered: one in which the internal density of the sphere is uniform and the other in which the density falls linearly from a central value to zero at the surface (the uniform and linear models). These assumptions served to define the distribution of angular momentum per unit mass with mass fraction. The spheres were flattened into disks, and models of the disks were found in which there was a force balance in the radial and vertical directions, subject to certain approximations, and with everywhere the assigned values of angular momentum per unit mass. The radial pressure gradient of the gas was included in the force balance. The energy transport in the vertical direction involved convection and radiative equilibrium; the principal contributors to opacity at lower temperatures were metallic iron grains and ice. The models contained two convection zones, an inner one due to the dissociation of hydrogen molecules, and an outer one in which there was a high opacity due to metallic iron grains. The characteristic semithickness of the disks ranged from about 0.1 astronomical units near the center to about one astronomical unit near the exterior. Characteristic angular momentum transport times and radiation lifetimes for these models of the initial solar nebula were estimated. Both types of characteristic lifetime were as short as a few years near the inner part of the models, and became about 10⁴ years or longer at distances greater than ten astronomical units.     

Experimental studies of magnetite formation in the solar nebula

Abstract— Oxidation of Fe metal and Gibeon meteorite metal to magnetite via the net reaction 3 Fe (metal) + 4 H₂O (gas) = Fe₃O₄ (magnetite) + 4 H₂ (gas) was experimentally studied at ambient atmospheric pressure at 91–442 °C in H₂ and H₂-He gas mixtures with H₂/H₂O molar ratios of ～4–41. The magnetite produced was identified by x-ray diffraction. Electron microprobe analyses showed 3.3 wt% NiO and 0.24 wt% CoO (presumably as NiFe₂O₄ and CoFe₂O₄) in magnetite formed from Gibeon metal. The NiO and CoO concentrations are higher than expected from equilibrium between metal and oxide under the experimental conditions. Elevated NiO contents in magnetite were also observed by metallurgists during initial stages of oxidation of Fe-Ni alloys. The rate constants for magnetite formation were calculated from the weight gain data using a constant surface area model and the Jander, Ginstling-Brounshtein, and Valensi-Carter models for powder reactions. Magnetite formation followed parabolic (i.e., diffusion-controlled) kinetics. The rate constants and apparent activation energies for Fe metal and Gibeon metal are: These rate constants are significantly smaller than the parabolic rate constants for FeS growth on Fe metal in H₂S-H₂ gas mixtures containing 1000 or 10 000 ppmv H₂S (Lauretta et al., 1996a). The experimental data for Fe and Gibeon metal are used to model the reaction time of Fe alloy grains in the solar nebula as a function of grain size and temperature. The reaction times for 0.1–1 μm radius metal grains are generally within estimated lifetimes of the solar nebula (0.1–10 Ma). However, the calculated reaction times are probably lower limits, and further study of magnetite formation at larger H₂/H₂O ratios, at lower temperatures and pressures, and as a function of metal alloy composition is needed for further modeling of nebular magnetite formation.     

Models of the formation of the solar nebula

Models of the collapse of a protostellar cloud and the formation of the solar nebula reveal that the size of the nebula produced will be the larger of $\text{R}{}_{\mathrm{<mtext>CF</mtext>}}\text{≡ J}{}^2\text{/k}{}^2\text{GM}{}^3\text{and}\text{R}{}_{\mathrm{<mtext>V</mtext>}}\text{≡ (GMv/2c}{}_{\mathrm{<mtext>c</mtext>}}{}^3\text{)}\text{1}\text{2}$ (where J, M, and c_s are the total angular momentum, total mass, and sound speed of the protosetellar material; G is the gravitational constant; k is a number of order unity; and v is the effective viscosity in the nebula). From this result it can be deduced that low-mass nebulas are produced if P ≡ (R_V/R_CF)² ? 1; “massive” nebulas result if P ? 1. Gravitational instabilities are expected to be important for the evolution of P ? 1 nebulas. The value of J distinguishes most current models of the solar nebula, since P ∝ J^?4. Analytic expressions for the surface density, nebular mass flux, and photospheric temperature distributions during the formation stage are presented for some simple models that illustrate the general properties of growing protostellar disks. It does not yet seem possible to rule out either P ? 1 or P < 1 for the solar nebula, but observed or possible heterogeneities in composition and angular-momentum orientation favor P < 1 models.     

The first ten million years in the solar nebula

Abstract— The ²⁶Al/²⁷Al ratio in a large number of calcium-aluminum inclusions (CAIs) is a rather uniform 5 × 10^?5, whereas in chondrules the ratio is either undetectable or has a much lower value; the simplest interpretation of this is that there was an interval of a few million years between the times that these two meteoritic constituents formed stable solids. The present investigation was undertaken as an exploration of the physics of the processes in the solar nebula during and after the accumulation of the Sun. Understanding the time scales of events in this nebular model, to see if this would cast light on this apparent CAI to chondrule time interval, was the major motivation for the exploration. There were four stages in the history of the solar nebula; in stage 1, a fragment of an interstellar molecular cloud collapsed to form the Sun and solar nebula; in stage 2, the nebula was in approximate steady state balance between infall from the cloud and accretion onto the Sun and was in its FU Orionis accumulation stage; in stage 3, the Sun had been mainly accumulated and there was a slow residual mass flow into the Sun while it was in its classical T Tauri stage; and in stage 4, the nebula had finished accreting material onto the Sun (now a weak-lined T Tauri star) and was in a static condition with no significant dissipation or motions, other than removal at the inner edge due to the T Tauri solar wind and photoevaporation beyond 9 astronomical units (AU). It is found that the energy source keeping the nebula warm during stages 3 and 4 is recombination of ionized H in the ionized bipolar jets and the T Tauri coronal expansion solar wind. The parameters of the heating model were adjusted to locate the ice sublimation line at 5.2 AU. In this work, a nebular model is used with a surface density of 4.25 × 10³ gm/cm² at 1 AU and a variation with radial distance as the inverse first power. Under normal conditions in the nebula, there is a negative pressure gradient that provides partial radial support for the gas, which thus circles the Sun more slowly than large solid objects do. Large objects undergo a slow inward spiral due to the gas drag; very small objects move essentially with the gas but have a slow inward drift; and intermediate objects (e.g., 1 m) have a fairly large inward drift velocity that traverses the full radial extent of the nebula in considerably less than the CAI to chondrule time interval. Such objects are thus lost unless they can grow rapidly to larger sizes. Near the inner edge (bow) of the nebula during stage 4, the pressure gradient becomes positive, creating a narrow zone of zero gas drag toward which solids drift from both directions, facilitating planetesimal formation in the inner solar nebula. Recent theoretical and experimental results on sticking probabilities of solids show that icy surfaces have the best sticking properties, but icy interstellar grains can only stick together when subjected to impact velocities of less than 2000 cm/sec. However, if the solid objects are very underdense, then a collision leads to interpenetration and many points at which the small constituent grains can adhere to one another, and thus coagulation becomes possible for such underdense objects. Simulations were made of such coagulation in the outer solar nebula, and it was found that the central plane of the nebula quickly becomes filled with meter-sized and larger bodies that rapidly accumulated near the top of the nebula and rapidly descended; in a few thousand years this quickly leads to gravitational instabilities that can form planetesimals. These processes led to the rapid formation of Jupiter in the nebula (and the slightly less rapid formation of the other giant planets). The early formation of Jupiter opens an annular gap in the nebula, and thus a second region is created in the nebula with zero gas drag. It is concluded that CAIs were formed at the end of stage 2 of the nebula history and moved out into the nebula for long-term storage, and that most chondrules were formed by magnetic reconnection flares in the bow region of the nebula during stage 4, several million years later. Carbonaceous meteorites should be formed on the far side of the Jovian gap, with the chondrules being heated by flares on the early Jupiter irradiating materials in the nearby zone of zero gas drag, and they should have essentially the same ²⁶Al ages as the CAIs (this will be very hard to confirm owing to scarcity of Al mineral phases in these chondrules).     

26Al and the partial ionization of the solar nebula

Calculation of the ionization state and consequent magnetic Reynolds number for the solar nebula shows that the presence of²⁶Al will result in strong coupling of the gas to magnetic fields. In the absence of²⁶Al,⁴⁰K will still result in substantial ionization, but the degree of magnetic coupling is much more model dependent.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.also Department of Astronomy.     

The distributions and ages of refractory objects in the solar nebula

Refractory objects such as Calcium, Aluminum-rich Inclusions, Amoeboid Olivine Aggregates, and crystalline silicates, are found in primitive bodies throughout our Solar System. It is believed that these objects formed in the hot, inner solar nebula and were redistributed during the mass and angular momentum transport that took place during its early evolution. The ages of these objects thus offer possible clues about the timing and duration of this transport. Here we study how the dynamics of these refractory objects in the evolving solar nebula affected the age distribution of the grains that were available to be incorporated into planetesimals throughout the Solar System. It is found that while the high temperatures and conditions needed to form these refractory objects may have persisted for millions of years, it is those objects that formed in the first 10⁵ years that dominate (make up over 90%) those that survive throughout most of the nebula. This is due to two effects: (1) the largest numbers of refractory grains are formed at this time period, as the disk is rapidly drained of mass during subsequent evolution and (2) the initially rapid spreading of the disk due to angular momentum transport helps preserve this early generation of grains as opposed to later generations. This implies that most refractory objects found in meteorites and comets formed in the first 10⁵ years after the nebula formed. As these objects contained live ²⁶Al, this constrains the time when short-lived radionuclides were introduced to the Solar System to no later than 10⁵ years after the nebula formed. Further, this implies that the t=0 as defined by meteoritic materials represents at most, the instant when the solar nebula finished accreting significant amounts of materials from its parent molecular cloud.     

The distribution of mass in the planetary system and solar nebula

A model solar nebula is constructed by adding the solar complement of light elements to each planet, using recent models of planetary compositions. Uncertainties in this approach are estimated. The computed surface density varies approximately asr ^–3/2. Mercury, Mars and the asteroid belt are anomalously low in mass, but processes exist which would preferentially remove matter from these regions. Planetary masses and compositions are generally consistent with a monotonic density distribution in the primordial solar nebula.