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.     

Solar wind origin of 36Ar on Venus

A hypothesis is considered in which the ³⁶Ar found on Venus is of solar origin. This possibility is quantitatively discussed within the framework of present theories of planetary accumulation by sweep up of planetesimals under gas-free conditions. Solar wind implantation of ³⁶Ar would take place by irradiation of accumulating material during the first ≈10⁵ years of planetary growth, provided that the flux of solar wind was enhanced by a factor of ≈100 at that time. Enrichment of Venus in implanted gas would be a consequence of the irradiated material being initially confined to the innermost edge of the radially opaque circusolar planetesimal disk predicted by these theories. The observed atmospheric data require a Ne/Ar fractionation by a factor of ≈100 during the planetesimal stage. It is also necessary that there be very little mixing of irradiated planetesimals from the inner edge of disk to the distance (≈1 AU) at which the Earth formed. The hypothesis can be tested by measurement of the abundance of Kr and Xe in the Venus atmosphere. Venera data indicate a terrestrial ³⁶Ar/Kr ratio, in disagreement with the solar wind hypothesis. In contrast, the Pioneer experiments find a lower limit to this ratio, well above the terrestrial value, that is compatible with the hypothesis. These experiments also show that Venus' ³⁶Ar/Xe ratio does not correspond to the so-called “planetary” trapped inert gas composition. The inert of Venus could be related to result of admixture of gas with solar composition. The inert gas on Venus could be related to that found in enstatite chondrites.     

The range of validity of the two-body approximation in models of terrestrial planet accumulation: II. Gravitational cross sections and runaway accretion

The validity of the two-body approximation in calculating encounters between planetesimals has been evaluated as a function of the ratio of unperturbed planetesimal velocity (with respect to a circular orbit) to mutual escape velocity when their surfaces are in contact (V/V_e). Impact rates as a function of V/V_e are calculated to within ～20% by numerical integration of the equations of motion. It is found that when V/V_e > 0.4, the two-body approximation is a good one. At low velocities (V/V_e < 0.1) two-body “collision-course” trajectories fail to lead to impacts. On the other hand, at these low velocities many impacts result from encounter trajectories with unperturbed separation distances far beyond the two-body gravitational radius. These two effects tend to cancel, and the resulting impact rates remain within a factor of ～3 of the two-body value in spite of these major differences in the nature of the impact trajectories. Therefore, on the average, the two-body approximation is useful well below the value of V/V_e for which it fails to describe individual encounters, and the required corrections are not large. As a consequence of this “anomalous gravitational focusing” planetesimals will continue to interact even when their orbits are noncrossing. This reduces the difficulty with premature isolation of planetesimal embryos during accumulation. Quantitatively, when 0.06 ? V/V_e ? 0.2, the impact rate varies approximately with the fifth power of the radius of the larger body, and is about a factor of 3 above that predicted using the conventional two-body gravitational cross-section formula. At lower values of V/V_e , the impact rate increases less rapidly. Finally, at the lowest values of V/V_e (<.02), the impact rate increases only in proportion to the geometric cross section, as a consequence of the swarm being essentially two dimensional for large unperturbed encounter distances. The gravitational enhancement in effective cross section is thereby limited to a value of about 3000. This leads to an optimal size for growth of planetesimals from a swarm of given eccentricity, and places a limit on the extent of runaway accretion.     

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.     

Ammonia photolysis and the greenhouse effect in the primordial atmosphere of the earth

Photochemical calculations indicate that in the prebiotic atmosphere of the Earth ammonia would have been irreversibly converted to N₂ in less than 40 years if the ammonia surface mixing ratio were ≤ 10^?4. However, if a continuous outgassing of ammonia were maintained, radiative equilibrium calculations indicate that a surface mixing ratio of ammonia of 10^?5 or greater would provide a sufficient greenhouse effect to keep the surface temperature above freezing. With a 10^?4 mixing ratio of ammonia, 60 to 70% of the present day solar luminosity would be adequate to maintain surface temperatures above freezing. A lower limit to the time constant for accumulation of an amount of nitrogen equivalent to the present day value is 10 my if the outgassing were such as to provide a continuous surface mixing ratio of ammonia ≥ 10^?5.     

Calculations of the evolution of the giant planets

Evolutionary calculations are presented for spherically symmetric protoplanetary configurations with a homogeneous solar composition and with masses of 10^?3, 1.5 × 10^?3, 2.85 × 10^?4, and 4.2 × 10^?4M_⊙. Recent improvements in equation-of-state and opacity calculations are incorporated. Sequences start as subcondensations in the solar nebula with densities of ～10^?10 to 10^?11 g cm^?3, evolve through a hydrostatic phase lasting 10⁵ to 10⁷ years, undergo dynamic collapse due to dissociation of molecular hydrogen, and regain hydrostatic equilibrium with densities ～1 g cm^?3. The nature of the objects at the onset of the final phase of cooling and contraction is discussed and compared with previous calculations.     

The evolution of an impact-generated atmosphere

Shock wave and thermodynamic data for rock-forming and volatile-bearing minerals are used to determine minimum impact velocities (v_cr) and minimum impact pressures (p_cr) required to form a primary H₂O atmosphere during planetary accretion from chondritelike planetesimals. The escape of initially released water from an accreting planet is controlled by the dehydration efficiency. Since different planetary surface porosities will result from formation of a regolith, v_cr and p_cr can vary from 1.5 to 5.8 km/sec and from 90 to 600 kbar, respectively, for target porosities between 0 and ～45%. On the basis of experimental data, hydration rates for forsterite and enstatite are derived. For a global regolith layer on the Earth's surface, the maximum hydration rate equals 6 × 10¹⁰ g H₂O sec^?1 during accretion of the Earth. Attenuation of impact-induced shock pressure is modeled to the extent that the amount of released water as a function of projectile radius, impact velocity, weight fraction of water in the target, target porosity, and dehydration efficiency can be estimated. The two primary processes considered are the impact release of water bound in hydrous minerals (e.g., serpentine) and the subsequent reincorporation of free water by hydration of forsterite and enstatite. These processes are described in terms of model calculations for the accretion of the Earth. Parameters which lead to a primary atmosphere/hydrosphere are: an accretion time of ? 1.6 × 10⁸years, the use of an accretion model defined by Weidenschilling (1974, 1976), a mean planetesimal radius of 0.5 km, a hydration rate of 6 × 10¹⁰ g H₂O sec^?1 inferred from a mean porosity of ～ 10% for the upper 1 km of the accreting Earth, and values for the dehydration efficiency, DE, of 0.55 and 0.07 for the maximum and minimum pressure decay model, respectively. Conditions which prohibit the formation of a primary atmosphere include an accretion time much longer than 1.6 × 10⁸ years, a hydration rate for forsterite and enstatite well in excess of 6 × 10¹⁰ g H₂O sec^?1, and a dehydration efficiency DE < 0.07. We conclude that the concept of dehydration efficiency is of dominant importance in determining the degree to which an accreting planet acquires an atmosphere during its formation.     

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.     

Formation of the lunar atmosphere

Measurements of⁴⁰Ar and helium made by the Apollo 17 lunar surface mass-spectrometer are used in the synthesis of atmospheric supply and loss mechanisms. The argon data indicate that about 8% of the⁴⁰Ar produced in the Moon due to decay of⁴⁰K is released to the atmosphere and subsequently lost. Variability of the atmospheric abundance of argon requires that the source be localized, probably in an unfractionated, partially molten core. If so, the radiogenic helium released with the argon amounts to 10% of the atmospheric helium supply. The total rate of helium escape from the Moon accounts for only 60% of the solar windα particle influx. This seems to require a nonthermal escape mechanism for trapped solar-wind gases, probably involving weathering of exposed soil grain surfaces by solar wind protons.     

The spectrum of titan in the far-infrared and microwave regions

The pressure-induced absorptions of gaseous nitrogen (N₂) and methane (CH₄) are computed on the basis of the collisional lineshape theory of G. Birnhaum and E.R. Cohen [Canad. J. Phys.54, 593–602 (1976)]. Laboratory data at 300 and 124°K for N₂ and at 296 and 195°K for CH₄ are used to determine the collisional time constant and their temperature dependence. The spectrum of Titan from the microwave to the far-infrared region (0.1–600 cm^?1) is then modeled using these opacities and a temperature profile of Titan's atmosphere derived from the Voyager 1 radio occultation experiment. The model atmosphere is composed of N₂ and CH₄, their relative proportions being determined by the vapor pressure law of CH₄. A model with gaseous opacity alone is ruled out by the far-infrared observations. An additional opacity, thought to be associated with a methane cloud, is confirmed. The effective temperature of Titan is estimated at T_e = 83.2 ± 1.4°K.     

On the Inclination Distribution of the Jovian Irregular Satellites

Irregular satellites—moons that occupy large orbits of significant eccentricity e and/or inclination I—circle each of the giant planets. The irregulars often extend close to the orbital stability limit, about 1/3-1/2 of the way to the edge of their planet's Hill sphere. The distant, elongated, and inclined orbits suggest capture, which presumably would give a random distribution of inclinations. Yet, no known irregulars have inclinations (relative to the ecliptic) between 47 and 141°.This paper shows that many high-I orbits are unstable due to secular solar perturbations. High-inclination orbits suffer appreciable periodic changes in eccentricity; large eccentricities can either drive particles with ∼70°<I<110° deep into the realm of the regular satellites (where collisions and scatterings are likely to remove them from planetocentric orbits on a timescale of 10⁷-10⁹ years) or expel them from the Hill sphere of the planet.By carrying out long-term (10⁹ years) orbital integrations for a variety of hypothetical satellites, we demonstrate that solar and planetary perturbations, by causing particles to strike (or to escape) their planet, considerably broaden this zone of avoidance. It grows to at least 55°<I<130° for orbits whose pericenters freely oscillate from 0 to 360°, while particles whose pericenters are locked at ±90° (Kozai mechanism) can remain for longer times.We estimate that the stable phase space (over 10 Myr) for satellites trapped in the Kozai resonance contains ∼10% of all stable orbits, suggesting the possible existence of a family of undiscovered objects at higher inclinations than those currently known.     

An estimation of the fluctuations in the extreme limb of the Sun

Detailed computations of synthetic solar limb curves are carried out for the purpose of estimating the effects of inhomogeneities in the solar atmosphere upon the observed limb position. Methods of determining the limb position given a solar limb curve are compared. The method of finding the locus of a fixed intensity level with respect to the average disk-center intensity at a given wavelength seems to be the most tractable definition to use on noise free data. It is found that limb fluctuations due to the solar 5-min p-mode oscillations produce a fluctuation in the limb height of about 6 km (0.008 arc sec) rms. Limb fluctuations due to granulation and chromospheric structure are much smaller. The wavelength dependence of the solar H^? opacity causes the height of the limb to increase by about 35 km between 400 and 850 nm, thus leading to a ‘limb reddening’ at the extreme limb of the Sun.     

Molecular gas species in the lunar atmosphere

There is good evidence for the existence of very small amounts of methane, ammonia and carbon dioxide in the very tenuous lunar atmosphere which consists primarily of the rare gases helium, neon and argon. All of these gases, except⁴⁰Ar, originate from solar wind particles which impinge on the lunar surface and are imbedded in the surface material. Here they may form molecules before being released into the atmosphere, or may be released directly, as is the case for rare gases. Evidence for the existence of the molecular gas species is based on the pre-dawn enhancement of the mass peaks attributable to these compounds in the data from the Apollo 17 Lunar Mass Spectrometer. Methane is the most abundant molecular gas but its concentration is exceedingly low, 1 × 10³ mol cm^?3, slightly less than³⁶Ar, whereas the solar wind flux of carbon is approximately 2000 times that of³⁶Ar. Several reasons are advanced for the very low concentration of methane in the lunar atmosphere.     

A semi-analytic model for oligarchic growth

A new semi-analytic model for the oligarchic growth phase of planetary accretion is developed. The model explicitly calculates damping and excitation of planetesimal eccentricities e and inclinations i due to gas drag and perturbations from embryos. The effects of planetesimal fragmentation, enhanced embryo capture cross sections due to atmospheres, inward planetesimal drift, and embryo-embryo collisions are also incorporated. In the early stages of oligarchic growth, embryos grow rapidly as e and i fall below their equilibrium values. The formation of planetesimal collision fragments also speeds up embryo growth as fragments have low-e, low-i orbits, thereby optimizing gravitational focussing. At later times, the presence of thick atmospheres captured from the nebula aids embryo growth by increasing their capture cross sections. Planetesimal drift due to gas drag can lead to substantial inward transport of solid material. However, inward drift is greatly reduced when embryo atmospheres are present, as the drift timescale is no longer short compared to the accretion timescale. Embryo-embryo collisions increase embryo growth rates by 50% compared to the case where growth is solely due to accretion of planetesimals. Formation of 0.1-Earth-mass protoplanets at 1 AU and 10-Earth-mass cores at 5 AU requires roughly 0.1 and 1 million years respectively, in a nebula where the local solid surface density is 7 g cm⁻² at each of these locations.     

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.     

Trapping and dynamical evolution of interplanetary dust particles in Earth’s quasi-satellite resonance

We used numerical simulations to model the orbital evolution of interplanetary dust particles (IDPs) evolving inward past Earth’s orbit under the influence of radiation pressure, Poynting–Robertson light drag (PR drag), solar wind drag, and gravitational perturbations from the planets. A series of β values (where β is the ratio of the force from radiation pressure to that of central gravity) were used ranging from 0.0025 up to 0.02. Assuming a composition consistent with astronomical silicate and a particle density of 2.5 g cm⁻³ these β values correspond to dust particle diameters ranging from 200 μm down to 25 μm. As the dust particle orbits decay past 1 AU between 4% (for β = 0.02, or 25 μm) and 40% (for β = 0.0025, or 200 μm) of the population became trapped in 1:1 co-orbital resonance with Earth. In addition to traditional horseshoe type co-orbitals, we found about a quarter of the co-orbital IDPs became trapped as so-called quasi-satellites. Quasi-satellite IDPs always remain relatively near to Earth (within 0.1–0.3 AU, or 10–30 Hill radii, R_H) and undergo two close-encounters with Earth each year. While resonant perturbations from Earth halt the decay in semi-major axis of quasi-satellite IDPs their orbital eccentricities continue to decrease under the influence of PR drag and solar wind drag, forcing the IDPs onto more Earth-like orbits. This has dramatic consequences for the relative velocity and distance of closest approach between Earth and the quasi-satellite IDPs. After 10⁴–10⁵ years in the quasi-satellite resonance dust particles are typically less than 10R_H from Earth and consistently coming within about 3R_H. In the late stages of evolution, as the dust particles are escaping the 1:1 resonance, quasi-satellite IDPs can have deep close-encounters with Earth significantly below R_H. Removing the effects of Earth’s gravitational acceleration reveals that encounter velocities (i.e., velocities “at infinity”) between quasi-satellite IDPs and Earth during these close-encounters are just a few hundred meters per second or slower, well below the average values of 2–4 km s⁻¹ for non-resonant Earth-crossing IDPs with similar initial orbits. These low encounter velocities lead to a factor of 10–100 increase in Earth’s gravitationally enhanced impact cross-section (σ_grav) for quasi-satellite IDPs compared to similar non-resonant IDPs. The enhancement in σ_grav between quasi-satellite IDPs and cometary Earth-crossing IDPs is even more pronounced, favoring accretion of quasi-satellite dust particles by a factor of 100–3000 over the cometary IDPs. This suggests that quasi-satellite dust particles may dominate the flux of large (25–200 μm) IDPs entering Earth’s atmosphere. Furthermore, because quasi-satellite trapping is known to be directly correlated with the host planet’s orbital eccentricity the accretion of quasi-satellite dust likely ebbs and flows on 10⁵ year time scales synchronized with Earth’s orbital evolution.     

Secular resonance,solar spin down,and the orbit of Mercury

A mechanism capable of accounting for the large mean eccentricity (0.175) and inclination (7°.2) of Mercury is discussed. Provided the gravitational field of the rapidly rotating primordial Sun had a sufficiently large second degree harmonic (i.e., J₂ ? order 10^?3), subsequent solar spin down would drive the orbit of Mercury through two secular resonances with Venus, one involving the precession of the line of apsides, the other one involving the regression of the nodal line. Resonance passage generates contributions to the eccentricity and inclination that are proportional to the square root of the characteristic solar spin down time. We find that an initial solar rotation l period of $\text{P ? 5}\text{1}\text{2}\text{hr}$ guarantees passage through resonance and that a spin down time of $\text{τ = Ω|dΩ/dt|}{}^{\mathrm{?1}}$ of order 10⁶ years could have produced the observed eccentricity and inclination. Such a primordial rotation rate is comparable to the measured rotations of very young stars and the spin down time appears consistent with the time scale derived for magnetic braking of the Sun's rotation by an intense solar wind during a T-Tauri stage of solar evolution.     

Accurate Determination of the TOA Solar Spectral NIR Irradiance Using a Primary Standard Source and the Bouguer–Langley Technique

We describe an instrument dedicated to measuring the top of atmosphere (TOA) solar spectral irradiance (SSI) in the near-infrared (NIR) between 600 nm and 2300 nm at a resolution of 10 nm. Ground-based measurements are performed through atmospheric NIR windows and the TOA SSI values are extrapolated using the Bouguer–Langley technique. The interest in this spectral range arises because it plays a main role in the Earth’s radiative budget and also because it is employed to validate models used in solar physics. Moreover, some differences were observed between recent ground-based and space-based instruments that take measurements in the NIR and the reference SOLSPEC(ATLAS3) spectrum. In the 1.6 μm region, the deviations vary from 6 % to 10 %. Our measuring system named IRSPERAD has been designed by Bentham (UK) and has been radiometrically characterized and absolutely calibrated against a blackbody at the Belgian Institute for Space Aeronomy and at the Physikalisch-Technische Bundesanstalt (Germany), respectively. A four-month measurement campaign was carried out at the Izaña Atmospheric Observatory (Canary Islands, 2367 m a.s.l.). A set of top-quality solar measurements was processed to obtain the TOA SSI in the NIR windows. We obtained an average standard uncertainty of 1 % for 0.8 μm<λ<2.3 μm. At 1.6 μm, corresponding to the minimum opacity of the solar photosphere, we obtained an irradiance of 234.31±1.29 mWm^?2?nm^?1. Between 1.6 μm and 2.3 μm, our measurements show a disagreement varying from 6 % to 8 % relative to ATLAS3, which is not explained by the declared standard uncertainties of the two experiments.     

Numerical experiment applicable to the latest stage of planet growth

A three-dimensional numerical model was developed with the goal of studying limited dynamical problems relevant to the latest stage of planet growth in the accretion theory. A small number of large protoplanets (～ Moon size) of different masses, moving around the Sun, are considered. The dynamical evolution and growth of the population is studied under mutual gravitational perturbations, accretion, and collisional fragmentation processes. Gravitational encounters are treated exactly by numerical integration of the N-body problem. Outcomes of collisional fragmentation are modeled according to the results of R. Greenberg et al. (1978, Icarus, 35, 1–26). In the present work, we consider 25 protoplanets with uniform mass distribution in the range 2 × 10²⁵?4 × 10²⁶ g on heliocentric orbits in the Earth zone. These bodies are initially confined to a small volume of space to permit gravitational perturbations by close approaches and collisions within a finite length of integration time. The dynamical evolution of the swarm is followed for four different sets of initial ranges in semimajor axis, eccentricity, and inclination: Δa=0.01, 0.02, 0.04, 0.08 AU; Δe= 0.005, 0.01, 0.02, 0.04; Δi=0°3, 0°6, 1°2, 2°4. Among other results, it is found that average eccentricities and inclinations evolve toward a steady state such that $\text{i}\text{?}\text{1}\text{2}\text{,}\text{e}$; it is also found that, whatever the initial conditions, the population evolves toward a quasi-equilibrium relative velocity distribution corresponding to a Safronov parameter value $\text{θ?10}$. Moreover, the growth process of the growing planet presents very similar behavior in the four cases considered, except for the time scale of evolution, which increases with the initial range of orbital elements. Earlier works of this kind have been presented by L.P. Cox and J.S. Lewis (1980, Icarus, 44, 706–721) and by G.N. Wetherill (1980b, In Geol. Soc. Canad. Spec. Publ., p. 20), although a number of differences exist between the three approaches.     

On the structure of Mars' atmosphere at 120–220 km

Altitude dependences of [CO₂] and [CO₂⁺] are deduced from Mariner 6 and 7 CO₂⁺ airglow measurements. CO₂ densities are also obtained from n_e radio occultation measurements. Both [CO₂] profiles are similar and correspond to the model atmosphere of Barth et al. (1972) at 120 km, but at higher altitudes they diverge and at 200–220 km the obtained [CO₂] values are three times less the model. Both the airglow and radio occultation observations show that a correction factor of 2.5 should be included into the values for solar ionization flux given by Hinteregger (1970). The ratio of [CO₂⁺]/n_e is 0.15–0.2 and, hence, [O]/[CO₂] is ～3% at 135 km. An atmospheric and ionospheric model is developed for 120–220 km. The calculated temperature profile is characterized by a value of T ≈ 370°K at h ? 220 km, a steep gradient (～2°/km) at 200-160 km, a bend in the profile at 160 km, a small gradient (～0.7°/km) below and a value of T ≈ 250°K at 120 km. The upper point agrees well with the results of the Lyman-α measurements; the steep gradient may be explained by molecular viscosity dissipation of gravity and acoustical waves (the corresponding energy flux is 4 × 10^?2 erg cm^?2sec^?1 at 180 km). The bend at 160 km may be caused by a sharp decrease of the eddy diffusion coefficient and defines K ≈ 2 × 10⁸cm²sec^?1; and the low gradient gives an estimate of the efficiency of the atmosphere heating by the solar radiation as ? ≈ 0.1.