Dynamic analysis by direct superposition of Ritz vectors

The solution of the eigenvalue problem for large structures is often the most costly phase of a dynamic response analysis. In this paper, the need for the exact solution of this large eigenvalue problem is eliminated. A new algorithm, based on error minimization, is presented for the generation of a sequence of Ritz vectors. These orthogonal vectors are used to reduce the size of the system. Only Ritz vectors with a large participation factor are used in the subsequent mode superposition analysis. In all examples studied, the superposition of Ritz vectors yields more accurate results, with fewer vectors, than if the exact eigenvectors are used. The proposed method not only reduces computer time requirements significantly but provides an error estimation for the dynamic analysis. The approach automatically includes the advantages of the proven numerical techniques of static condensation, Guyan reduction and static correction due to higher mode truncation.     

Extinct I129 in C3 chondrites

Eight C3 chondrites were examined by the I¹²⁹Xe¹²⁹ dating method, to see whether their IXe “ages” (better, initial $\text{I}{}^{129}\text{I}{}^{127}$ratios ≡ R₀) correlate with any other properties. The R₀'s range from 1.60 × 10^?4 to 1.09 × 10^?4, corresponding to IXe ages from 2.0 Myr before to 6.7 Myr after Murchison magnetite. Three C3O's (Lancé, Felix, Ornans) have essentially indistinguishable R₀'s of (1.41 ± 0.13) to (1.17 ± 0.10) × 10^?4; the fourth C3O, Warrenton, is undatable owing to homogenization of radiogenic and trapped Xe.Four C3V's show a distinct spread: Vigarano and Grosnaja are highest [R₀ = (1.60 ± 0.07) and (1.57 ± 0.14) × 10^?4], Mokoia is intermediate, and Kaba is lowest [R₀ = (1.38 ± 0.06) and (1.09 ± 0.10) × 10^?4]. Literature values for Allende place it near Kaba. These R₀'s correlate inversely with 4 other properties: I-, Br-, and Cd-content, and olivine composition, both percent mean deviation (PMD) and proportion of iron-poor olivine grains (≤2% fayalite).It is difficult to accept the ～9 Myr spread in R₀ as a true age, reflecting either nebular or parent-body processes. This time span is more than an order of magnitude longer than the lifetime of the solar nebula inferred from astronomical evidence. Nor does the degree of thermal metamorphism, which is slight for C3's anyway, correlate with R₀. A more plausible interpretation is that the variations in R₀ reflect mainly isotopic heterogeneity of iodine. The simplest model that accounts for the correlations with R₀ involves mixing of two iodine components in the solar nebula, associated with gas and grains, respectively. The second, of lower $\text{I}{}^{129}\text{I}{}^{127}$ ratio, predominated at later times and thus became enriched in late-formed meteorites, along with other volatiles such as Cd and Br. The low Fe content and large PMD of olivine may reflect either less metamorphism owing to shallow location in the parent body, or greater reduction of Fe²⁺ during chondrule formation.     

The speciation of water in silicate melts

Previous models of water solubility in silicate melts generally assume essentially complete reaction of water molecules to hydroxyl groups. In this paper a new model is proposed that is based on the hypothesis that the observed concentrations of molecular water and hydroxyl groups in hydrous silicate glasses reflect those of the melts from which they were quenched. The new model relates the proportions of molecular water and hydroxyl groups in melts via the following reaction describing the homogeneous equilibrium between melt species: H₂O_molecular (melt) + oxygen (melt) = 2OH (melt). An equilibrium constant has been formulated for this reaction and species are assumed to mix ideally. Given an equilibrium constant for this reaction of 0.1–0.3, the proposed model can account for variations in the concentrations of molecular water and hydroxyl groups in melts as functions of the total dissolved water content that are similar to those observed in glasses. The solubility of molecular water in melt is described by the following reaction: H₂O (vapor) = H₂O_molecular (melt).These reactions describing the homogeneous and heterogeneous equilibria of hydrous silicate melts can account for the following observations: the linearity between f_H2O and the square of the mole fraction of dissolved water at low total water contents and deviations from linearity at high total water contents; the difference between the partial molar volume of water in melts at low total water contents and at high total water contents; the similarity between water contents of vapor-saturated melts of significantly different compositions at high pressures versus the dependence on melt composition of water solubility in silicate melts at low pressures; and the variations of viscosity, electrical conductivity, the diffusivity of “water,” the diffusivity of cesium, and phase relationships with the total dissolved water contents of melts.This model is thus consistent with available observations on hydrous melt systems and available data on the species concentrations of hydrous glasses and is easily tested, since measurements of the concentrations of molecular water and hydroxyl groups in silicate glasses quenched from melts equilibrated over a range of conditions and total dissolved water contents are readily obtainable.     

Crystallization sequences of Ca-Al-rich inclusions from Allende: An experimental study

The equilibrium crystallization sequence at 1 atmosphere in air of a melt corresponding in composition to the average composition of Type B Ca-Al-rich inclusions from the Allende meteorite is: spinel (1550°C) → melilite (1400°C; Åk22) → anorthite (1260°C) → Ti-Al-rich clinopyroxene (1230°C; “Ti-fassaite”). The melilite becomes increasingly åkermanitic with decreasing temperature. The pyroxene is similar in composition to fassaites from Type B inclusions. Preliminary results suggest that the crystallization sequence is similar at oxygen fugacities near the iron-wüstite buffer.The results of these experiments have been integrated with available phase equilibrium data in the system CaO-MgO-Al₂O₃-SiO₂TiO₂ and a phase diagram for predicting the crystallization sequences of liquids with compositions of coarse-grained Ca-Al-rich inclusions has been developed.Available bulk compositions of coarse-grained inclusions form a well-defined trend in terms of major elements, extending from Type A and Bl inclusions near the spinel-melilite join to more pyroxene-rich Type B2 inclusions. The trend deviates from the expected sequence of solid condensates from a nebular gas at P = 10^?3 atm if pure diopside is assumed to be the clinopyroxene that condenses. The Type A-B1 end of the trend is similar in composition to calculated equilibrium condensates at 1202–1227°C and the trend as a whole parallels the sequence of condensates expected from diopside condensation at ~ 1170°C. The trend is consistent to first order with the condensation of solid Ti-rich fassaite in place of pure diopside at higher temperatures than those at which pure diopside is predicted to condense. Partially molten condensates may be likely in this case or if the nebular pressure is higher than 10^?3 atm.     

On the strength of electric currents and zonal magnetic fields at the top of the Earth''s core: Methodology and preliminary estimates

A new method is described for estimating: (a) the meridional electric current density, j_θ, (b) the vertical growth rate of the zonal magnetic field, ∂B_φ/∂r, or its scale-height, B_φ/∂B_φ/∂r) and (c) the vertical growth rate of the vertical current density, ∂j_r/∂r, at a few isolated points on the top surface of the Earth's core from observations of the internal geomagnetic field at the Earth's surface. The theoretical technique rests on combining unaccelerated, gravitationally-driven Boussinesq fluid dynamics of the core with frozen-flux electromagnetism, the mantle being treated as a spherically symmetric insulator.

Insertion into this theory of main field models for epochs 1965, 1975 leads to preliminary values for these quantities of magnitude: (a) j_θ 1 A/m², (b) ∂B_φ/∂r 10⁻⁶ T/m or B_φ/(∂B_φ/∂r) 10 m, (c) ∂j_r/∂r 10⁻⁶ A/m³. Some geophysical implications of these estimates are discussed.     

Trace elements in shergottite meteorites: Implications for the origins of planets

The average concentrations of 19 siderophile and volatile elements in shergottite meteorites differ from those in terrestrial basalts by less than a factor of ten. This observation undermines claims that the abundances of siderophile and volatile elements in the Earth's upper mantle are uniquely terrestrial. Claims that similarities in the Moon's siderophile element pattern imply a terrestrial origin for the Moon are also weakened. The implication that basalt source regions on the asteroidal parent body of the shergottites resembled the terrestrial upper mantle constrains models of planetary formation and evolution. Heterogeneous accretion models may explain many of the similarities between these planets. Alternatively, separation of sulfide from basaltic magmas or their source regions on the Earth and the shergottite parent body may explain some of these similarities.     

Isotopic anomalies of noble gases in meteorites and their origins—IV. C3 (Ornans) carbonaceous chondrites

The C3O chondrites Kainsaz, Lancé and Ornans were studied by an acid dissolution technique, to characterize the noble-gas components in 3 mineral fractions: HF, HCl-solubles (99% of the meteorite), chromite and carbon (0.3–0.9%), and ‘phase Q’, a poorly characterized trace mineral (0.05–0.4%) containing most of the Ar, Kr, Xe. For all fractions, gas contents decline in the order Kainsaz > Lancé > Ornans; this trend parallels volatile contents but not heterogeneity of olivine composition or degree of metamorphism and seems to reflect progressively higher condensation temperatures from the solar nebula.Solubles contain nearly unfractionated Xe, and show ${}^{136}\text{Ar}{}^{132}\text{Xe}$ ratios up to 850. Hence the high $\text{Ar}\text{Xe}$ ratios (200–400) of bulk C3O chondrites must be due to an HF-soluble mineral (possibly magnetite). Phase Q contains ordinary planetary gases and a Ne component of ${}^{20}\text{Ne}{}^{22}\text{Ne}\text{= 10.3 ± 0.4.}$Chromite and carbon contain Ne of ${}^{20}\text{Ne}{}^{22}\text{Ne}\text{= 8.6 ± 0.1}$ and ‘CCF’ xenon (a peculiar component of possibly fissiogenic origin, enriched in the heavy isotopes but accompanied by a component enriched in the light isotopes).In all primitive chondrites, both the amount and the chemical separability of CCFXe parallel the abundance of promordial noble gases and other volatiles, such as C, N, Tl, Bi and In. The close correlation of CCFXe with various properties of undoubtedly local origin (volatile content, petrologic type, presence of ferrichromite and carbon, etc.) is more consistent with a local than with an extrasolar origin of this component. A volatile superheavy element seems to be the most plausible source, but the evidence is not conclusive.