Non-radial stellar oscillations: a perspective from potential scattering (I)

This article is the first in a series designed to gain insight into the stellar oscillation problem from a somewhat novel point of view: that of potential scattering, well-known in the quantum mechanical literature. In this paper the known theoretical foundations are developed and applied in the context of the astrophysical problem, wherein the star itself (rather than any portion of it) is the potential which scatters waves and traps them. The basis for the identification of a precisely defined scattering problem is the existence of a linear Schrödinger equation associated both globally (Section 2) and locally (Section 8) with the nonlinear eigenvalue equation for nonradial stellar pulsations. The paper is also designed to be a fairly complete account of the relevant mathematical topics that are germane to a study of this kind. This paper is dedicated to the memory of Professor Zdenèk Kopal, who was a great source of professional encouragement to me during the last fifteen years of his life.     

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.     

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.     

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 .     

Age — heaping: contrasting patterns from china

Detailed examination of China's 1982 census data confirms that the standard of age reporting among the Han Chinese, and among many of the ethnic minorities, is very good. Furthermore the quality of age reporting is unaffected by large variations in levels of literacy which exist between men and women, between young and old, and between rural and urban areas. Much of the credit for the high quality of China's age reporting stems from the distinctive nature of Chinese traditional calendars. However, in the far west of China, in the Xinjiang Autonomous Region, age heaping and age exaggeration are as bad as they are in many other developing countries. The authors show that the poor quality of age reporting in Xinjiang is confined to three of the region's ethnic communities, the Uygur, Kirgiz and Tajik, who appear to neither know, nor care, how old they are.     

IGU commission on the world political map

The International Geographical Union Reports

IGU commission on the world political map     

Surface-layer flow in complex terrain: Comparison of models and full-scale observations

Four models of surface boundary-layer flow in complex terrain are compared with observations made at Blashaval Hill, North Uist, Scotland. The field experiment is described by Mason and King (1985). Three of the models are derived from the two-dimensional theory of Jackson and Hunt (1975) and are described in Mason and King (1985), Walmsley et al. (1986) and Troen and Petersen (1989). The fourth is a mass-consistent code based on Traci et al. (1979). The model results are in good agreement with each other and are generally within the observed range of variation ( ~ ± 16%) in normalized wind speed. For most wind direcions (7 of 11), model results of normalized wind speed at the summit were within 7% of the observed mean values. For some wind directions, calculations using the Guidelines of Walmsley et al. (1989) suggested that variations in surface roughness were important. This led us to apply one of our models incorporating nonuniform surface roughness. The lack of significant improvement for cases when water lay upstream of Blashaval Hill is attributed to compensating changes at summit and reference sites and to very local effects on the wind data. Sensitivity to topography lying to the west and northwest of Blashaval was also investigated. Results suggested an influence from those distant topographic features for some wind directions. When those features were incorporated, maximum errors in normalized wind speed at the summit were reduced from 18 to 13%.     

Deep drilling at the Siljan Ring impact structure: oxygen-isotope geochemistry of granite

The Siljan Ring is a 362-Ma-old impact structure formed in 1700-Ma-old I-type granites. A 6.8-km-deep borehole provides a vertical profile through granites and isolated horizontal diabase sills. Fluid-inclusion thermometry, and oxygen-isotope compositions of vein quartz, granite, diabase, impact melt, and pseudotachylite, reveal a complex history of fluid activity in the Siljan Ring, much of which can be related to the meteorite impact. In granites from the deep borehole, ¹⁸O values of matrix quartz increase with depth from near 8.0 at the surface to 9.5 at 5760 m depth. In contrast, feldspar ¹⁸O values decrease with depth from near 10 at the surface to 7.1 at 5760 m, forming a pattern opposite to the one defined by quartz isotopic compositions. Values of ¹⁸O for surface granites outside the impact structure are distinct from those in near-surface samples from the deep borehole. In the deep borehole, feldspar coloration varies from brick-red at the surface to white at 5760 m, and the abundances of crack-healing calcite and other secondary minerals decrease over the same interval. Superimposed on the overall decrease in alteration intensity with depth are localized fracture zones at 4662, 5415, and 6044 m depth that contain altered granites, and which provided pathways for deep penetration of surface water. The antithetic variation of quartz and feldspar ¹⁸O values, which can be correlated with mineralogical evidence of alteration, provides evidence for interaction between rocks and impact-heated fluids (100–300° C) in the upper 2 km of the pluton. Penetration of water to depths below 2 km was restricted by a general decrease in impact-fracturing with depth, and by a 60-m-thick diabase sill at 1500 m depth that may have been an aquitard. At depths below 4 km in the pluton, where water/rock ratios were low, oxygen isotopic compositions preserve evidence for limited high-temperature (>500° C) exchange between alkali feldspar and fluids. The high-temperature exchange may have been a post-impact event involving impact-heated fluids, or a post-magmatic event.     

Comparison of crude oils and their alteration products

A simple method is proposed for using gas Chromatographic data on the high molecular weight saturated fractions of crude oils to make comparisons of source materials, migration mechanisms and, in particular, alteration processes within reservoirs. The preferential loss of normal alkanes by bacterial attack on a light crude is one of several processes readily followed using this scheme.