Data from the Global Oscillation Network Group (GONG) project and other helioseismic experiments provide a test for models of stellar interiors and for the thermodynamic and radiative properties, on which the models depend, of matter under the extreme conditions found in the sun. Current models are in agreement with the helioseismic inferences, which suggests, for example, that the disagreement between the predicted and observed fluxes of neutrinos from the sun is not caused by errors in the models. However, the GONG data reveal subtle errors in the models, such as an excess in sound speed just beneath the convection zone. These discrepancies indicate effects that have so far not been correctly accounted for; for example, it is plausible that the sound-speed differences reflect weak mixing in stellar interiors, of potential importance to the overall evolution of stars and ultimately to estimates of the age of the galaxy based on stellar evolution calculations.

https://science.sciencemag.org/content/sci/272/5266/1286.full.pdf

The Current State of Solar Modeling

We calculate accurate solar models and report the detailed time dependences of important solar quantities. We use helioseismology to constrain the luminosity evolution of the Sun and report the discovery of semiconvection in evolved solar models that include diffusion. In addition, we compare the computed sound speeds with the results of p-mode observations by BiSON, GOLF, GONG, LOWL, and MDI instruments. We contrast the neutrino predictions from a set of eight standard-like solar models and four deviant (or deficient) solar models with the results of solar neutrino experiments. For solar neutrino and helioseismological applications, we present present-epoch numerical tabulations of characteristics of the standard solar model as a function of solar radius, including the principal physical and composition variables, sound speeds, neutrino fluxes, and functions needed for calculating solar neutrino oscillations.

https://iopscience.iop.org/article/10.1086/321493/pdf

Solar Models: Current Epoch and Time Dependences, Neutrinos, and Helioseismological Properties

Helium and heavy-element diffusion are both included in precise calculations of solar models. In addition, improvements in the input data for solar interior models are described for nuclear reaction rates, the solar luminosity, the solar age, heavy-element abundances, radiative opacities, helium and metal diffusion rates, and neutrino interaction cross sections. The effects on the neutrino fluxes of each change in the input physics are evaluated separately by constructing a series of solar models with one additional improvement added at each stage. The effective 1 σ uncertainties in the individual input quantities are estimated and used to evaluate the uncertainties in the calculated neutrino fluxes and the calculated event rates for solar neutrino experiments. The calculated neutrino event rates, including all of the improvements, are 9.3-1.4+1.2 SNU for the 37Cl experiment and 137-7+8 SNU for the 71Ga experiments. The calculated flux of 7Be neutrinos is 5.1 (1.00-0.07+0.06)×10^9 cm^-2 s^-1 and the flux of 8B neutrinos is 6.6(1.00-0.17+0.14)×10^6 cm^-2 s^-1. The primordial helium abundance found for this model is Y=0.278. The present-day surface abundance of the model is Ys=0.247, in agreement with the helioseismological measurement of Ys=0.242±0.003 determined by Hernandez and Christensen-Dalsgaard (1994). The computed depth of the convective zone is R=0.712R⊙, in agreement with the observed value determined from p-mode oscillation data of R=0.713±0.003R⊙ found by Christensen-Dalsgaard et al. (1991). Although the present results increase the predicted event rate in the four operating solar neutrino experiments by almost 1 σ (theoretical uncertainty), they only slightly increase the difficulty of explaining the existing experiments with standard physics (i.e., by assuming that nothing happens to the neutrinos after they are created in the center of the sun). For an extreme model in which all diffusion (helium and heavy-element diffusion) is neglected, the event rates are 7.0-1.0+0.9 SNU for the 37Cl experiment and 126-6+6 SNU for the 71Ga experiments, while the 7Be and 8B neutrino fluxes are, respectively, 4.5(1.00-0.07+0.06)×10^9 cm^-2 s^-1 and 4.9(1.00-0.17+0.14)×10^6 cm^-2 s^-1. For the no-diffusion model, the computed value of the depth of the convective zone is R=0.726R⊙, which disagrees with the observed helioseismological value. The calculated surface abundance of helium, Ys=0.268, is also in disagreement with the p-mode measurement. The authors conclude that helioseismology provides strong evidence for element diffusion and therefore for the somewhat larger solar neutrino event rates calculated in this paper.

/pdf/solar-models-with-helium-and-heavy-element-diffusion-1n6za700do.pdf

Solar models with helium and heavy-element diffusion

Helioseismology and Solar Abundances

The code CESAM is a consistent set of programs and routines which performs calculations of 1D quasi-static stellar evolution including diffusion and rotation. The principal new feature is the solution of the quasi-static equilibrium by collocation method based on piecewise polynomials approximations projected on their B-spline basis; that allows stable and robust calculations and the exact restitution of the solution not only at grid points. Another advantage is the monitoring by only one parameter of the accuracy and its improvement by superconvergence. An automatic mesh refinement has been designed for adjusting the localizations of grid points according to the changes of unknowns, each limit between a radiative and a convective zones being shifted to the closest grid point. For standard models, the evolution of the chemical composition is solved by stiffly stable schemes of orders up to four; for non-standard models the solution of the diffusion equation employs the Petrov-Galerkin scheme, with the mixing of chemicals in convective zones performed by strong turbulent diffusion. A precise restoration of the atmosphere is allowed for. CESAM computes evolution of stars from the pre-main sequence to the beginning of the 4 He burning cycle. In this paper a detailed description of the algorithms is presented.

/pdf/cesam-a-code-for-stellar-evolution-calculations-503518gadh.pdf

CESAM: A code for stellar evolution calculations

Data on the evolution of a solar model are presented which include the effects of diffusive processes that lead to a partial sorting of the elements with depth. The two models are with and without the effects of diffusion. The diffusion process of gravitational settling, composition gradient diffusion, and thermal diffusion are based on the method of Iben and MacDonald (1985). It is found that a high original helium content is required to match the observed global oscillation data, and that the total neutrino flux detectable by the chlorine detector accordingly is increased to over 10 SNU. 111 refs.

/pdf/oscillations-of-solar-models-with-internal-element-diffusion-8xalbiappi.pdf

Oscillations of Solar Models with Internal Element Diffusion

Two precision solar models have been constructed with the Iben evolution program, one with no diffusion of the internal atomic nuclei, and another that includes the effects of gravitational settling, thermal diffusion, and concentration gradient diffusion on the element abundances. This paper reports the results of these models. 8 figs., 3 tabs.

/pdf/oscillations-of-solar-models-with-internal-element-diffusion-3oa0m26vz6.pdf

Oscillations of solar models with internal element diffusion

We have calculated periods and growth rates for a new class of nonradically pulsated variable stars in the linear, nonradial nonadiabatic theory. The best observed member of this class is PG 1159-035 which is observed to be pulsating in a number of periods around 500 s and may well have an effective temperature exceeding 10/sup 5/ K. Stellar envelopes constructed for a 0.6 M star are in the effective temperature range 7 x 10/sup 4/ K< or =T/sub e/< or =1.5 x 10/sup 5/ K following the white dwarf cooling track at radii 0.016-0.030 R. These envelopes have compositions of half carbon and half oxygen by mass. We assume that the observed periods are nonradial g-modes (l = 2 or 3) and search in the interval between approx.25 s and 1200 s, to find regions of instability for a very broad range of effective temperatures. Blue edges of our instability strips are strongly dependent upon both the thermal structure and the interior composition. Pure carbon envelopes have nonradial blue edge at T< or =10/sup 5/ K, while the half-carbon, half-oxygen envelopes have a nonradial blue edge exceeding 1.5 x 10/sup 5/ K. The pulsation excitation mechanism is the cyclical ionizationmore » of the K-shell electrons of carbon and oxygen at an interior temperature approx.10/sup 6/ K. These results imply that PG 1159-035 has a significant amount of oxygen present in its surface layers, and while current stellar evolution theory predicts that this oxygen is produced during helium shell flashes, its presence this close to the surface (M/sub s/approx. 10/sup 12/M)) implies that extremely efficient stellar mass loss mechanisms have ejected the entire hydrogen and most of the helium envelopes.« less

Nonradial instability strips based on carbon and oxygen partial ionization in hot, evolved stars

We have performed a linear, nonradial, nonadiabatic pulsation stability analysis of stellar models with effective temperatures of 1.50 x 10/sup 5/ K, 1.55 x 10/sup 5/ K, 1.60 x 10/sup 5/ K, and 1.65 x 10/sup 5/ K, and luminosities characteristic of stars evolving from the AGB to the white dwarf cooling sequence. Our purpose is to find unstable pulsation modes with periods as long as those observed in the central star of the planetary nebula, K1-16. These models have masses of 0.6 M/sub sun/ and homogeneous compositions of half-carbon and half-oxygen by mass. All these stellar models exhibit unstable nonradial modes with periods ranging from 1000 to 4000 s. The longest periods were found in the model with T/sub e/ = 1.55 x 10/sup 5/ K, L = 2000 L/sub sun/. The pulsation driving mechanism is the cyclical ionization of carbon and oxygen at temperatures of (2--5) x 10/sup 6/ K. This analysis demonstrates that K1-16 is probably a member of the PG 1159-035 class of sdO variable stars. This star is, therefore, an evolutionary link from the central stars of the OVI planetary nebulae to the hottest non-Da white dwarfs as proposed by Sion, Liebert, and Starrfield. Itmore » must also be evolving very rapidly.« less

An analysis of nonradial pulsations of the central star of the planetary nebula K1-16.

In order to see if there could be agreement between results of stellar evolution theory and those of nonradial pulsation theory, calculations of white dwarf models have been made for hydrogen surface masses of 10−4 M ⊙. Earlier results by Winget et al. (1982) indicated that surface masses greater than 10−8 M ⊙ would not allow nonradial pulsations, even though all the driving and damping is in surface layers only 10−12 of the mass thick. We show that the surface mass of hydrogen in the pulsating white dwarfs (ZZ Ceti variables) can be any value as long as it is thick enough to contain the surface convection zone.

/pdf/pulsations-of-white-dwarf-stars-with-thick-hydrogen-or-11ov7l5m2g.pdf

Russell B. Kidman

Papers

Oscillations of Solar Models with Internal Element Diffusion

Oscillations of solar models with internal element diffusion

Nonradial instability strips based on carbon and oxygen partial ionization in hot, evolved stars

An analysis of nonradial pulsations of the central star of the planetary nebula K1-16.

Pulsations of white dwarf stars with thick hydrogen or helium surface layers