Fundamental properties of solar-like oscillating stars from frequencies of minimum $Delta u$ : II. Model computations for different chemical compositions and mass

The large separations between the oscillation frequencies of solar-like stars are measures of stellar mean density. The separations have been thought to be mostly constant in the observed range of frequencies. However, detailed investigation shows that they are not constant, and their variations are not random but have very strong diagnostic potential for our understanding of stellar structure and evolution. In this regard, frequencies of the minimum large separation are very useful tools. From these frequencies, in addition to the large separation and frequency of maximum amplitude, Yi ldi z et al. recently have developed new methods to find almost all the fundamental stellar properties. In the present study, we aim to find metallicity and helium abundances from the frequencies, and generalize the relations given by Yi ldi z et al. for a wider stellar mass range and arbitrary metallicity ($Z$) and helium abundance ($Y$). We show that the effect of metallicity is { significant} for most of the fundamental parameters. For stellar mass, for example, the expression must be multiplied by $(Z/Z_{sun})^{0.12}$. For arbitrary helium abundance, $ M propto (Y/Y_{sun})^{0.25} $. Methods for determination of $Z$ and $Y$ from pure asteroseismic quantities are based on amplitudes (differences between maximum and minimum values of Dnu) in the oscillatory component in the spacing of oscillation frequencies. Additionally, we demonstrate that the difference between the first maximum and the second minimum is very sensitive to $Z$. It also depends on $ u_{rm min1}/ u_{rm max}$ and small separation between the frequencies. Such a dependence leads us to develop a method to find $Z$ (and $Y$) from oscillation frequencies. The maximum difference between the estimated and model $Z$ values is about 14 per cent. It is 10 per cent for $Y$.