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On the distance of the globular cluster M4 (NGC 6121) using RR Lyrae stars: I. optical and near-infrared Period-Luminosity and Period-Wesenheit relations

We present new distance determinations to the nearby globular M4 (NGC~6121) based on accurate optical and Near Infrared (NIR) mean magnitudes for fundamental (FU) and first overtone (FO) RR Lyrae variables (RRLs), and new empirical optical and NIR Period-Luminosity (PL) and Period-Wesenheit (PW) relations. We have found that optical-NIR and NIR PL and PW relations are affected by smaller standard deviations than optical relations. The difference is the consequence of a steady decrease in the intrinsic spread of cluster RRL apparent magnitudes at fixed period as longer wavelengths are considered. The weighted mean visual apparent magnitude of 44 cluster RRLs is $\left<V\right>=13.329\pm0.001$ (standard error of the mean) $\pm$0.177 (weighted standard deviation) mag. Distances were estimated using RR Lyr itself to fix the zero-point of the empirical PL and PW relations. Using the entire sample (FU$+$FO) we found weighted mean true distance moduli of 11.35$\pm$0.03$\pm$0.05 mag and 11.32$\pm$0.02$\pm$0.07 mag. Distances were also evaluated using predicted metallicity dependent PLZ and PWZ relations. We found weighted mean true distance moduli of 11.283$\pm$0.010$\pm$0.018 mag (NIR PLZ) and 11.272$\pm$0.005$\pm$0.019 mag (optical--NIR and NIR PWZ). The above weighted mean true distance moduli agree within 1$σ$. The same result is found from distances based on PWZ relations in which the color index is independent of the adopted magnitude (11.272$\pm$0.004$\pm$0.013 mag). These distances agree quite well with the geometric distance provided by \citep{kaluzny2013} based on three eclipsing binaries. The available evidence indicates that this approach can provide distances to globulars hosting RRLs with a precision better than 2--3\%.