Kiyoshi Nakazawa

The Gas Drag Effect on the Elliptic Motion of a Solid Body in the Primordial Solar Nebula

The gas drag effect in the primordial solar nebula on the motion of a solid body, ranging from a large planetesimal to a small dust grain, is investigated. For a planetesimal, expressions for the short-term variation of the semi-major axis, the eccentricity and inclination are obtained using a perturbation method and a realistic formula for the gas drag force. For a small body, an orbit spiralling towards the sun in the equatorial plane of the nebula is investigated and the decay time of the orbital radius is obtained. It is found that the decay time at 1 a.u. is as short as 10 or 10 years for a solid body with mass in the range between 10 g and 10 g.

Earth's melting due to the blanketing effect of the primordial dense atmosphere

When the proto-Earth was growing by the accretion of planetesimals and its mass became greater than about 0.1 ME, where ME is the present Earths mass, an appreciable amount of gas of the surrounding solar nebula was attracted towards the proto-Earth to form an optically thick, dense atmosphere. We have studied the structure of this primordial atmosphere under the assumptions that (1) it is spherically symmetric and in hydrostatic equilibrium, and (2) the net energy outflow (i.e., the luminosity) is constant throughout the atmosphere and is given by GMM/R with M = M/106yr or M/107yr where M and R are the mass and the radius of the proto-Earth, respectively. n nThe results of calculations show that the temperature at the bottom of the atmosphere, namely, at the surface of the proto-Earth increases greatly with the mass of the proto-Earth and it is about 1500°K for M = 0.25 ME. This high temperature is due to the blanketing effect of the opaque atmosphere. Thus, as long as the primordial solar nebula was existing, the surface temperature of the proto-Earth was kept high enough to melt most of the materials and, hence, the melted iron sedimented towards the center to form the Earths core.

Growth and sedimentation of dust grains in the primordial solar nebula

Abstract Of the formation processes in the solar system, the process of growth and sedimentation of dust grains in the primordial solar nebula is investigated for a region near the Earths orbit. The growth equation for dust grains, which are sinking as well as being in thermal motion, is solved numerically in the wide mass range between 10 −12 and 10 6 g. Any turbulent motions in the nebula are assumed to have already decayed when the sedimentation begins. The numerical simulation shows that the growth and sedimentation proceed faster than was found by Kusaka et al. (1970) but in accordance with the estimate of Safronov (1969) owing to a cooperative interaction of the growth and the sedimentation; that is, at about 3 × 10 3 years after the beginning of the growth and sedimentation a dust layer, composed of centimeter-sized grains, is formed at the equator of the solar nebula. Furthermore, the mass density of dust grains floating in the outer layers of the nebula is found to be of the order of 10 −5 after 10 5 years compared with that before the sedimentation. From these results, it can be estimated that at about 5 × 10 3 years after the beginning of sedimentation the dust layer breaks up owing to the onset of gravitational instability.

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.

Dissolution of the primordial rare gases into the molten Earth's material

Abstract We have shown in a previous paper that, if the primordial solar nebula existed when the Earth was formed, the Earth was once surrounded by a dense and massive primordial atmosphere, whose temperature and pressure were about 4000 K and 900 atm, respectively, at the bottom. We suppose that this hydrogen-rich atmosphere escaped from the Earth after the solar nebula itself disappeared, both phenomena probably being due to the effect of strong solar wind and radiation. Using the results of our previous and new calculations on the structure of the primordial atmosphere, we have investigated the amount of dissolution of the rare gases, which were contained in the primordial atmosphere, into the molten Earths material. The amount of the dissolved rare gases is found to be strongly dependent on the grain opacity of the atmosphere, i.e., on the amount of fine grains. However, their isotopic ratios and relative abundance are independent of the opacity and approximately equal to those in the primordial solar nebula, that is, to the present solar values. Especially, the dissolved neon is expected to have remained in the present mantle. Therefore, if a considerable amount of neon with nearly the solar isotopic ratio is discovered in present mantle material, this offers direct evidence for the proposition that the proto-Earth was once surrounded by the primordial atmosphere.

Dissipation of the Primordial Terrestrial Atmosphere Due to irradiation of the Solar EUV

The escape of a primordial Earths atmosphere due to heating by solar radiation is studied by integrating numerically hydrodynamic equations for steady and sphericallyxad symmetric outflow of hydrogen molecules and helium atoms. As heating sources, we take account of (1) the solar EUV radiation which is expected to be very strong during the T Tauri stage, (2) the solar visible light and (3) the release of gravitational energy of accreting planetesimals. The effect of solar wind is neglected but the condition of this neglect is estimated. The results show that the primordial atmosphere, having existed in the early stage of the Earths history, is dissipated within a period of 5 X 108 y, which is the upper limit imposed from the origin of the present terrestrial atmosphere, as far as the solar EUV flux is more than 2 >< 10 2 times as large as the present one.

Dissipation of the rare gases contained in the primordial Earth's atmosphere

Abstract If the Earth was formed by accumulation of rocky bodies in the presence of the gases of the primordial solar nebula, the Earth at this formation stage was surrounded by a massive primordial atmosphere (of about 1 × 1026 g) composed mainly of H2 and He. We suppose that the H2 and He escaped from the Earth, owing to the effects of strong solar wind and EUV radiation, in stages after the solar nebula itself dissipated into the outer space. The primordial atmosphere also contained the rare gases Ne, Ar, Kr and Xe whose amounts were much greater than those contained in the present Earths atmosphere. Thus, we have studied in this paper the dissipation of these rare gases due to the drag effect of outflowing hydrogen molecules. By means of the two-component gas kinetic theory and under the assumption of spherically symmetric flow, we have found that the outflow velocity of each rare gas relative to that of hydrogen is expressed in terms of only two parameters — the rate of hydrogen mass flow across the spherical surface under consideration and the temperature at this surface. According to this result, the rare gases were dissipated below the levels of their contents in the present atmosphere, when the mass loss rate of hydrogen was much greater than 1 × 1017 g/yr throughout the stages where the atmospheric mass decreased from 1 × 1026 g to 4 × 1019 g.

The Earth's core formation due to the Rayleigh-Taylor instability

Abstract The recent theories of planetary formation lead to a gravitationally unstable structure of the proto-Earth in the accretion stage, which is composed of three layers: an innermost undifferentiated solid core, an intermediate metal-melt layer, and an outermost silicate-melt layer. Taking this configuration as an initial state, we investigate the Earths core formation due to the Rayleigh-Taylor instability by using the quantitative results on the instability in a self-gravitating fluid sphere obtained from another paper (S. Ida, Y. Nakagawa, and K. Nakagawa, submitted). We find that the instability occurs through the translational mode on a time scale of about 10 hr if the thickness of the metal-melt layer ⪆1 km. This leads to the conclusion that the Earths core began to form through the translation of the innermost undifferentiated solid core as soon as the outer layer was melted and differentiated in the late accretion stage. In addition, we examine the rotational effects of the instability; the translation occurs most often along the rotational axis. But this preference is weak, since the rotational energy is small compared to the gravitational one.

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

Abstract 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.

Origin of the Moon — Capture by gas drag of the Earth's primordial atmosphere

We propose a new scenario of the lunar origin, which is a natural extension of planetary formation processes studied so far by us in Kyoto. According to these studies, the Earth grew up in a gaseous solar nebula and, consequently, the sphere of its gravitational influence (i.e., the Hill sphere of the Earth) was filled by a gas forming a dense primordial atmosphere of the Earth. In the later stages, this atmosphere as well as the solar nebula was dissipated gradually, owing to strong activities of the early-Sun in a T Tauri-stage.In the present and the subsequent papers, we study a series of dynamical processes where a lowenergy (i.e., slightly unbound) planetisimal is trapped within the terrestrial Hill sphere, under the above-mentioned circumstances that the gas density of the primordial atmosphere is gradually decreasing. It is clear that two conditions must be satisfied for the lunar origin: first, an unbound planetesimal entering the Hill sphere have to dissipate its kinetic energy and come into a bound orbit before it escapes from the Hill sphere and, second, the bound planetisimal never falls onto the surface of the Earth.In this paper we study the first condition by calculating the oribital motion of a planetesimal in the Hill sphere, which is affected both by solar gravity and by atmospheric gas drag. The results show that a low-energy planetisimal with the lunar mass or less can be trapped in the Hill sphere with a high probability, if it enters the Hill sphere at stages before the atmospheric density is decreased to about 1/50 of the initial value.In the subsequent paper, the second condition will be studied and it will be shown that a tidal force, among other forces, is very important for a trapped planetesimal to avoid collision with the Earth and stay eternally in the Hill sphere as a satellite.