On the Use of Field RR Lyrae as Galactic Probes. II. A new $\Delta$S
  calibration to estimate their metallicity
  We performed the largest and most homogeneous spectroscopic survey of field
RR Lyraes (RRLs). We secured $\approx$6,300 high resolution (HR, R$\sim$35,000)
spectra for 143 RRLs (111 fundamental, RRab; 32 first overtone, RRc). The
atmospheric parameters were estimated by using the traditional approach and the
iron abundances were measured by using an LTE line analysis. The resulting iron
distribution shows a well defined metal-rich tail approaching solar iron
abundance. This suggests that field RRLs experienced a complex chemical
enrichment in the early halo formation. We used these data to develop a new
calibration of the $\Delta$S method. This diagnostic, based on the equivalent
widths of CaII K and three Balmer (H$_{\delta,\gamma,\beta}$) lines, traces the
metallicity of RRLs. For the first time the new empirical calibration: i)
includes spectra collected over the entire pulsation cycle; ii) includes RRc
variables; iii) relies on spectroscopic calibrators covering more than three
dex in iron abundance; iv) provides independent calibrations based on
one/two/three Balmer lines. The new calibrations were applied to both
SEGUE-SDSS and degraded HR spectra totalling 6,451 low resolution (LR,
R$\sim$2,000) spectra for 5,001 RRLs (3,439 RRab, 1,562 RRc). This resulted in
an iron distribution with a median of -1.55$\pm$0.01 and $\sigma$=0.51 dex, in
good agreement with literature values. We also found that RRc are 0.10 dex more
metal-poor than RRab variables, and have a distribution with a smoother
metal-poor tail. This finding supports theoretical prescriptions suggesting a
steady decrease in the RRc number when moving from metal-poor to metal-rich
stellar environments.
