Metallicity of Galactic RR Lyrae from Optical and Infrared Light Curves:
  II. Period-Fourier-Metallicity Relations for First Overtone RR Lyrae
  We present new period-$\phi_{31}$-[Fe/H] relations for first overtone RRL
stars (RRc), calibrated over a broad range of metallicities ($-2.5 <
\textrm{[Fe/H]}< 0.0$) utilizing the largest currently available set of
Galactic halo field RRL with homogeneous spectroscopic metallicities. Our
relations are defined in the optical (ASAS-SN $V$-band) and, inaugurally, in
the infrared (WISE $W1$ and $W2$ bands). Our $V$-band relation can reproduce
individual RRc spectroscopic metallicities with a dispersion of 0.30 dex over
the entire metallicity range of our calibrator sample (an RMS smaller than what
we found for other relations in literature including non-linear terms). Our
infrared relation has a similar dispersion in the low and intermediate
metallicity range ($\textrm{[Fe/H]} < -0.5$) but tends to underestimate the
[Fe/H] abundance around solar metallicity. We tested our relations by measuring
both the metallicity of the Sculptor dSph and a sample of Galactic globular
clusters, rich in both RRc and RRab stars. The average metallicity we obtain
for the combined RRL sample in each cluster is within $\pm 0.08$ dex of their
spectroscopic metallicities. The infrared and optical relations presented in
this work will enable deriving reliable photometric RRL metallicities in
conditions where spectroscopic measurements are not feasible; e.g., in distant
galaxies or reddened regions (observed with upcoming Extremely Large Telescopes
and the James Webb Space Telescope), or in the large sample of new RRL that
will be discovered in large-area time-domain photometric surveys (such as LSST
and the Roman space telescope).
