Fundamental properties of Kepler and CoRoT targets – III. Tuning scaling relations using the first adiabatic exponent

So-called scaling relations based on oscillation frequencies have the potential to reveal the mass and radius of solar-like oscillating stars. In the derivation of these relations, it is assumed that the first adiabatic exponent at the surface ( $\Gamma _{\rm \negthinspace 1s}$ ) of such stars is constant. However, by constructing interior models for the mass range 0.8–1.6 M⊙, we show that $\Gamma _{\rm \negthinspace 1s}$ is not constant at stellar surfaces for the effective temperature range with which we deal. Furthermore, the well-known relation between large separation and mean density also depends on $\Gamma _{\rm \negthinspace 1s}$ . Such knowledge is the basis for our aim of modifying the scaling relations. There are significant differences between masses and radii found from modified and conventional scaling relations. However, a comparison of predictions of these relations with the non-asteroseismic observations of Procyon A reveals that new scaling relations are effective in determining the mass and radius of stars. In the present study, solar-like oscillation frequencies of 89 target stars (mostly Kepler and CoRoT) were analysed. As well as two new reference frequencies (νmin1 and νmin2) found in the spacing of solar-like oscillation frequencies of stellar interior models, we also take into account νmin0. In addition to the frequency of maximum amplitude, these frequencies have a very strong diagnostic potential in the determination of fundamental properties. The present study applies the derived relations from the models to the solar-like oscillating stars, and computes their effective temperatures using purely asteroseismic methods. There are in general very close agreements between effective temperatures from asteroseismic and non-asteroseismic (spectral and photometric) methods. For the Sun and Procyon A, for example, the agreement is almost total.