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Context. Asteroseismology is an efficient tool not only for testing stellar structure and evolutionary theory but also constraining the parameters of stars for which solar-like oscillations are presently detected. As an important southern asteroseismic target τ Ceti, is a metal-poor star. The main features of the oscillations and some frequencies of τ Ceti have been identified. Many scientists propose to comprehensively observe this star as part of the Stellar Observations Network Group. Aims. Our goal is to obtain the optimal model and reliable fundamental parameters for the metal-poor star τ Ceti by combining all non-asteroseismic observations with these seismological data. Methods. Using the Yale stellar evolution code (YREC), a grid of stellar model candidates that fall within all the error boxes in the HR diagram have been constructed, and both the model frequencies and large-and small-frequency separations are calculated using the Guenther's stellar pulsation code. The χ 2 νc minimization is performed to identify the optimal modelling parameters that reproduce the observations within their errors. The frequency corrections of near-surface effects to the calculated frequencies using the empirical law, as proposed by Kjeldsen and coworkers, are applied to the models. Results. We derive optimal models, corresponding to masses of about 0.775-0.785 M and ages of about 8-10 Gyr. Furthermore, we find that the quantities derived from the non-asteroseismic observations (effective temperature and luminosity) acquired spectroscopically are more accurate than those inferred from interferometry for τ Ceti, because our optimal models are in the error boxes B and C, which are derived from spectroscopy results.

Aims. Element diffusion is a basic physical element transport mechanism which induces the redistribution of chemical elements. Using the asteroseismic method, we study the effects of helium and heavy element diffusion on the internal structure and stellar evolution of solar-type stars. We also provide asteroseismic parameters for a grid of models which would be useful for direct comparison with the Kepler mission observations. Methods. We construct a grid of solar-type stellar models with various masses (from 0.8 M to 1.2 M ) and metallicities (Z i = 0.03, 0.025, 0.02, 0.015, 0.01, 0.005) with and without helium and heavy element diffusion. We compute "second differences" and "small separations" of the solar-metallicity models (Z i = 0.02) to analyze the effects of diffusion on the convection zone, helium abundance and the evolutionary sequence of the star. In order to study the asteroseismic property of models with and without diffusion, we compute the p-mode oscillation frequencies of low-degree modes for a grid of models and construct the ( Δν 0 , d 02 ) asteroseismic diagram. Results. We find that the element diffusion could speed up the evolution of the star, especially in the main sequence. The results show that it could enlarge the convective core and change the base of the convection envelope. In addition, the helium and heavy element diffusion make the models evolve to lower large and small separations in the asteroseismic diagram. This effect is more efficient at lower metallicity.

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The monthly sunspot numbers compiled by Temmer et al. and the monthly polar faculae from observations of the National Astronomical Observatory of Japan, for the interval of March 1954 to March 1996, are used to investigate the phase relationship between polar faculae and sunspot activity for total solar disk and for both hemispheres in solar cycles 19, 20, 21 and 22. We found that (1) the polar faculae begin earlier than sunspot activity, and the phase difference exhibits a consistent behaviour for different hemispheres in each of the solar cycles, implying that this phenomenon should not be regarded as a stochastic fluctuation;(2) the inverse correlation between polar faculae and sunspot numbers is not only a long-term behaviour, but also exists in short time range; (3) the polar faculae show leads of about 50-71 months relative to sunspot numbers, and the phase difference between them varies with solar cycle; (4) the phase difference value in the northern hemisphere differs from that in the southern hemisphere in a solar cycle, which means that phase difference also existed between the two hemispheres. Moreover, the phase difference between the two hemispheres exhibits a periodical behaviour. Our results seem to support the finding of Hiremath (2010).