On a New Method to Estimate the Distance, Reddening, and Metallicity of RR Lyrae Stars Using Optical/Near-infrared (B, V, I, J, H, K) Mean Magnitudes: ω Centauri as a First Test Case

We developed a new approach to provide accurate estimates of metal content, reddening and true distance modulus of RR Lyrae stars (RRLs). The method is based on homogeneous optical ($BVI$) and near-infrared ($JHK$) mean magnitudes and on predicted period--luminosity--metallicity relations ($IJHK$) and absolute mean magnitude--metallicity relations ($BV$). We obtained solutions for three different RRL samples in $\omega$ Cen: first overtone (RRc,~90), fundamental (RRab,~80) and global (RRc+RRab) in which the period of first overtones were fundamentalized. The metallicity distribution shows a well defined peak at [Fe/H]$\sim$--1.98 and a standard deviation of $\sigma$=0.54 dex. The spread is, as expected, metal-poor ([Fe/H]$\le$--2.3) objects. The current metallicity distribution is $\sim$0.3 dex more metal-poor than similar estimates for RRLs available in the literature. The difference vanishes if the true distance modulus we estimated is offset by --0.06/--0.07~mag in true distance modulus. We also found a cluster true distance modulus of $\mu$=13.720$\pm$0.002$\pm$0.030~mag, where the former error is the error on the mean and the latter is the standard deviation. Moreover, we found a cluster reddening of E($B-V$)=0.132$\pm$0.002$\pm$0.028~mag and spatial variations of the order of a few arcmin across the body of the cluster. Both the true distance modulus and the reddening are slightly larger than similar estimates available in the literature, but the difference is within 1$\sigma$. The metallicity dependence of distance diagnostics agree with theory and observations, but firm constraints require accurate and homogeneous spectroscopic measurements.