Asteroid motion near the 3 : 1 resonance

A mapping model is constructed to describe asteroid motion near the 3 : 1 mean motion resonance with Jupiter, in the plane. The topology of the phase space of this mapping coincides with that of the real system, which is considered to be the elliptic restricted three body problem with the Sun and Jupiter as primaries. This model is valid for all values of the eccentricity. This is achieved by the introduction of a correcting term to the averaged Hamiltonian which is valid for small values of the ecentricity.We start with a two dimensional mapping which represents the circular restricted three body problem. This provides the basic framework for the complete model, but cannot explain the generation of a gap in the distribution of the asteroids at this resonance. The next approximation is a four dimensional mapping, corresponding to the elliptic restricted problem. It is found that chaotic regions exist near the 3 : 1 resonance, due to the interaction between the two degrees of freedom, for initial conditions close to a critical curve of the circular model. As a consequence of the chaotic motion, the eccentricity of the asteroid jumps to high values and close encounters with Mars and even Earth may occur, thus generating a gap. It is found that the generation of chaos depends also on the phase (i.e. the angles andv) and as a consequence, there exist islands of ordered motion inside the sea of chaotic motion near the 3 : 1 resonance. Thus, the model of the elliptic restricted three body problem cannot explain completely the generation of a gap, although the density in the distribution of the asteroids will be much less than far from the resonance. Finally, we take into account the effect of the gravitational attraction of Saturn on Jupiter's orbit, and in particular the variation of the eccentricity and the argument of perihelion. This generates a mixing of the phases and as a consequence the whole phase space near the 3 : 1 resonance becomes chaotic. This chaotic zone is in good agreement with the observations.     

Primodial retention of carbon by the terrestrial planets

Chemical equilibrium calculations on the stability of pure and dissolved graphite and cohenite (Fe₃C), several other carbides, and several carbonates have been carried out for a system with solar elemental abundances over a very wide range of temperature and pressure. The calculated abundances of condensed carbon compounds are similar to the observed inventories on Earth and Venus, but fully 10 times smaller than the minimum carbon abundance found in ordinary chondrites. The total carbon content of most iron meteorites is compatible with their origin as a cooling FeNiCSP solution which was saturated with dissolved carbon at the solidus, such as would be produced by melting an ordinary chondrite, not by direct condensation from or equilibrium with the primitive solar nebula. It is argued that the carbon content of Mars need not be appreciably greater than that of the Earth. Material with even lower formation temperatures than Mars, such as the primitive material in the asteroid belt, may retain substantially more carbon as disequilibrium polymeric organic matter, possibly by the Fischer-Tropsch mechanism favored by Anders. Carbonates are not found as equilibrium products in a solar-composition system, and are probably secondary alteration products. CaCO₃ might, however, persist in a solar-composition gas at temperatures below 460°K and pressures below 10^?6.6 bar. The most stable condensed carbon compounds are found to be graphite, Fe₃C, and possibly TiC, all in solid solution in the metal phase.     

The condensation and fractionation of refractory lithophile elements

It has long been recognized that Cr, Mg, and Si are fractionated in chondritic material along with, but to a much lesser extent than, a large group of more refractory elements. Reasoning that this might imply some unique distribution at the time of fractionation, the patterns have been reexamined. It now appears as if two distinct fractionation patterns can be resolved: one involving ordinary and enstatite chondrites and the other involving carbonaceous chondrites, the Earth, the Moon, and the eucrite parent body. Significantly, the two trends inevitably intersect at C1 composition. Ordinary and enstatite chondrites appear to have evolved from C1 composition via the removal of about 40 and 56% of a high-temperature condensate. Another high-temperature condensate, with a distinctly different composition, appears to be enriched in the carbonaceous chondrites, the Moon, and possibly the Earth, but depleted in the eucrite parent body. The compositions of these two components are constrained to fall on the appropriate mixing lines. These lines intersect the condensation path at two points, one where Mg₂SiO₄ has just begun to condense (～20%) and a second where Mg₂SiO₄ was almost completely condensed (～90%). This represents about an 80° temperature difference. But it is within this range that the largest fraction of planetary matter (Mg, Si, and Fe) condenses. Conceivably the relatively sudden appearance of large amounts of condensed material is in some way related to the fractionation process, although the exact relationship cannot be specified.     

Thermodynamics of selected trace elements in the Jovian atmosphere

The thermochemistry of several hundred compounds of twelve selected trace elements (Ge, Se, Ga, As, Te, Pb, Sn, Cd, Sb, Tl, In, and Bi) has been investigated for solar composition material along a Jupiter adiabat. The results indicate that AsF₃, InBr, TlI, and SbS, in addition to CO, PH₃, GeH₄, AsH₃, H₂Se, HCl, HF, and H₃BO₃ proposed by Barshay and Lewis (1978), may be potential chemical tracers of atmospheric dynamics. The reported observations of GeH₄ is interpreted on the basis of new calculations as implying rapid vertical transport from levels where T ? 800°K. Upper limits are also set on the abundances of many gaseous compounds of the elements investigated.     

Mass-radius relationships and constraints on the composition of Pluto,II

A new estimate of Pluto's mass within the range of possible masses considered in an earlier work has enabled us to refine our model of Pluto's interior.     

Numerical simulation of the final stages of terrestrial planet formation

Three representative numerical simulations of the growth of the terrestrial planets by accretion of large protoplanets are presented. The mass and relative-velocity distributions of the bodies in these simulations are free to evolve simultaneously in response to close gravitational encounters and occasional collisions between bodies. The collisions between bodies, therefore, arise in a natural way and the assumption of expressions for the relative velocity distribution and the gravitational collision cross section is unnecessary. These simulations indicate that the growth of bodies with final masses approaching those of Venus and the Earth is possible, at least for the case of a two-dimensional system. Simulations assuming an initial uniform distribution of orbital eccentricities on the interval from 0 to e_max are found to produce final states containing too many bodies with masses which are too small when e_max < 0.10, while simulations with e_max > 0.20 result in too many catastrophic collisions between bodies thus preventing rapid accretion of planetary-size bodies. The e_max = 0.15 simulation ends with a state surprisingly similar to that of the present terrestrial planets and, therefore, provides a rough estimate of the range of radial sampling to be expected for the terrestrial planets.     

Newtonian cosmology with a varying gravitational constant

Newtonian cosmology is developed with the assumption that the gravitational constantG diminishes with time. The functional form adopted forG(t), a modification of a suggestion of Dirac, isG=A(k+t) ^–1, wheret is the age of the Universe and a small constantk is inserted to avoid a singularity in the two-body problem. IfR is the scale factor, normalized to unity at an epoch time , the differential equation is then . Here ₀ is the mean density at the epoch time. With the above form forG(t), the solution is reducible to quadratures.The scale factorR either increases indefinitely or has one and only one maximum. LetH ₀ be the present value of Hubble's constant /R and _0c the minimum density for a maximum ofR, i.e., for closure of the Universe. The conditions for a maximum lead to a boundary curve of _0c versusH ₀ and the numbers indicate strongly that thisG-variable Newtonian model corresponds to an open universe. An upward estimate of the age of the Universe from 10¹⁰ yr to five times such a value would still lead to the same conclusion.The present Newtonian cosmology appears to refute the statement, sometimes made, that the Dirac model forG necessarily leads to the conclusion that the age of the Universe is one-third the Hubble time. Appendix B treats this point, explaining that this incorrect conclusion arises from using all the assumptions in Dirac (1938). The present paper uses only Dirac's final result, viz,G(k+t)^–1, superposing it on the differential equation .