Context. In recent years the global seismic scaling relations for the frequency of maximum power, $\nu_{\mathrm max} \propto g/\sqrt{T_{\mathrm eff}}$, and for the large frequency separation, ${{\mathrm \Delta}}\nu \propto \sqrt{\bar\rho}$, have drawn attention in various fields of astrophysics. This is because these relations can be used to estimate parameters, such as the mass and radius of stars that show solar-like oscillations. With the exquisite photometry of Kepler, the uncertainties in the seismic observables are small enough to estimate masses and radii with a precision of only a few per cent. Even though this seems to work quite well for main-sequence stars, there is empirical evidence, mainly from studies of eclipsing binary systems, that the seismic scaling relations systematically overestimate the mass and radius of red giants by about 15% and 5%, respectively. Various model-based corrections of the Δν-scaling reduce the problem but do not solve it.
Aims. Our goal is to define revised seismic scaling relations that account for the known systematic mass and radius discrepancies in a completely model-independent way.
Methods. We use probabilistic methods to analyse the seismic data and to derive non-linear scaling relations based on a sample of six red giant branch (RGB) stars that are members of eclipsing binary systems and about 60 red giants on the RGB as well as in the core-helium burning red clump (RC) in the two open clusters NGC 6791 and NGC 6819.
Results. We re-examine the global oscillation parameters of the giants in the binary systems in order to determine their seismic fundamental parameters and we find them to agree with the dynamic parameters from the literature if we adopt non-linear scalings. We note that a curvature and glitch corrected Δνcor should be preferred over a local or average value of Δν. We then compare the observed seismic parameters of the cluster giants to those scaled from independent measurements and find the same non-linear behaviour as for the eclipsing binaries. Our final proposed scaling relations are based on both samples and cover a broad range of evolutionary stages from RGB to RC stars: $g/\sqrt{T_{\mathrm eff}} = (\nu_{\mathrm max}/\nu_{\mathrm max,\odot})^{1.0075\pm0.0021}$ and $\sqrt{\bar\rho} = ({{\mathrm \Delta}}\nu_{\mathrm cor}/{{\mathrm \Delta}}\nu_{\mathrm cor,\odot})[\eta - (0.0085\pm0.0025) \log^2 ({{\mathrm \Delta}}\nu_{\mathrm cor}/{{\mathrm \Delta}}\nu_{\mathrm cor,\odot})]^{-1}$, where g, Teff, and $\bar\rho$ are in solar units, νmax,⊙ = 3140 ± 5 μHz and Δνcor,⊙ = 135.08 ± 0.02 μHz, and η is equal to one in the case of RGB stars and 1.04 ± 0.01 for RC stars.
Conclusions. A direct consequence of these new scaling relations is that the average mass of stars on the ascending giant branch reduces to 1.10 ± 0.03 M⊙ in NGC 6791 and 1.45 ± 0.06 M⊙ in NGC 6819, allowing us to revise the clusters’ distance modulus to 13.11 ± 0.03 and 11.91 ± 0.03 mag, respectively. We also find strong evidence that both clusters are significantly older than concluded from previous seismic investigations

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