Asteroseismic Study on Cluster Distance Moduli for RGB Stars in NGC 6791 and NGC 6819

Stellar distance is an important basic parameter in stellar astrophysics. Stars in a cluster are thought to be formed coevally from the same interstellar cloud of gas and dust. They are therefore expected to have common properties. These common properties strengthen our ability to constrain theoretical models and/or to determine fundamental parameters, such as stellar mass, metal fraction, and distance when tested against an ensemble of cluster stars. Here we derive a new relation based on solar-like oscillations, photometric observations, and the theory of stellar structure and evolution of red giant branch stars to determine cluster distance moduli through the global oscillation parameters $\Delta\nu$ and $\nu_{\rm max}$, and photometric data \textit{V}. The values of $\Delta\nu$ and $\nu_{\rm max}$ are derived from \textit{kepler} observations. At the same time, it is used to interpret the trends between \textit{V} and $\Delta\nu$. From the analyses of this newly derived relation and observational data of NGC 6791 and NGC 6819 we devise a method in which all stars in a cluster are regarded as one entity to determine the cluster distance modulus. This approach fully reflects the characteristic of member stars in a cluster as a natural sample. From this method we derive true distance moduli of $13.09\pm0.10$ mag for NGC 6791 and $11.88\pm0.14$ mag for NGC 6819. Additionally, we find that the distance modulus only slightly depends on the metallicity [Fe/H] in the new relation. A change of 0.1 dex in [Fe/H] will lead to a change of 0.06 mag in the distance modulus.