A model of an exploding,radiating star in general relativity

In a previous paper Adams, Cary and Cohen (1994) presented a model of a supernova. In that paper the equations of General Relativity describing the evolution of a spherically symmetric, radiating star were solved analytically. The evolution of the star was determined by the application of boundary conditions at the center and at the edge. Due to lmitations in the presupernova model, only the very slow inward motion of an unstable, degenerate core could be considered. The solution was also limited by the need to exclude a runaway term, one that increased exponentially with time. Without the exclusion of the runaway, the luminosity would have increased without bound and the mass would have become negative.This paper presents a completely analytic solution to the equations of General Relativity describing the evolution of a Type II supernova. Professor S.E. Woosley kindly gave us data on the physical variables of a 12M ₀ presupernova star. In our model the core collapses within 1 s, leaving a 1.3M ₀ remnant. Shortly afterward 10.6M ₀ is ejected to infinity, and 0.17M ₀ is radiated away in the form of neutrinos. The distance of the edge from the center increases proportionally to the two-thirds power of the time. The luminosity decreases proportionally to the inverse four-thirds power.Although the runaway solution was modified by the exploding rather than a static envelope, it must still be excluded by adjusting initial conditions. Its character is changed from an exponential to a very large power (55) of time. The removal of a degree of freedom by this exclusion leads to physically non-sensical results such as negative luminosity. The inclusion of a term describing motion of the mantle due to neutrino interactions provides the additional degree of freedom necessary for physically reasonable results.     

The relationship between zooplankton,conductivity and lake-water ionic composition in 111 lakes from the Interior Plateau of British Columbia,Canada

Zooplankton collected from vertical net tows were related to the environmental variables from 98 lakes from the Interior Plateau of British Columbia. Canonical correspondence analysis showed that both salinity and ionic composition (pH and Mg) of the lake-water made major and significant contributions to the first two ordination axes (=0.42 and 0.11 respectively,P<0.05). BothArtemia franciscana andMoina hutchinsoni had their highest relative abundance in meso-hypersaline waters. However,Artemia franciscana preferred waters that were higher in Mg and Ca, whileMoina hutchinsoni was found in waters that were lower in Mg and Ca. Similarly, at intermediate salinities,Daphnia pulex and the calanoid copepods preferred waters slightly lower in Mg and Ca, whereasCeriodaphnia laticaudata andSimocephalus spp. were relatively more common in waters higher in Mg and Ca. Because the freshest lakes studied varied much less in ionic composition, the zooplankton in these lakes did not show a preference to ionic composition. As expected, multi-generic groups, such as the calanoid copepods, cyclopoid copepods and nauplii, had wider tolerances to conductivity than groups identified to lower taxonomic levels. Significant weighted-averaging regression and calibration models of conductivity were developed based on zooplankton species composition from the study lakes (r ²=0.56,P<0.05). Samples composed largely of multi-generic taxa yielded the worst estimates of salinity in the reconstruction model. This study suggests that zooplankton community composition may be developed into a useful proxy for paleosalinity reconstruction.     

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.     

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.     

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 .