  So called scaling relations have the potential to reveal the mass and radius
of solar-like oscillating stars, based on oscillation frequencies. In
derivation of these relations, it is assumed that the first adiabatic exponent
at the surface (Gamma_1s) of such stars is constant. However, by constructing
interior models for the mass range 0.8-1.6 Msun, we show that Gamma_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_1s. Such knowledge is the basis for our aim of
modifying scaling relations. There are significant differences between masses
and radii found from modified and conventional scaling relations. However,
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 (nu_min1 and nu_min2) found
in the spacing of solar-like oscillation frequencies of stellar interior
models, we also take into account nu_min0. In addition to the frequency of
maximum amplitude, these frequencies have very strong diagnostic potential for
determination of fundamental properties. The present study involves the
application of 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.
