Multi-line assessment of narrow-line regions in $z \sim$ 3 radio
  galaxies
  In this paper, we utilize high-quality rest-UV spectra of three radio
galaxies at $z \sim$ 3 observed with VLT/FORS2 to measure the flux of several
emission lines including relatively faint ones, such as NIV]$\lambda$1486,
OIII]$\lambda$1663, and [NeIV]$\lambda$2424. Additionally, we collect fluxes of
faint rest-UV emission lines in 12 $z \sim$ 3 radio galaxies from the
literature. Previously, physical and chemical properties of narrow-line regions
(NLRs) in high-$z$ active galactic nuclei (AGNs) have been investigated mostly
by using only strong rest-UV emission-lines (e.g., NV$\lambda$1240,
CIV$\lambda$1549, HeII$\lambda$1640, and CIII]$\lambda$1909). Such strong-line
diagnostics are based on various assumptions due to the limitation in the
number of available emission-line constraints. In this work, both physical and
chemical properties of NLR clouds in each object are estimated by fitting
detailed photoionization models to the measured emission-line fluxes. We
confirm that the metallicity of NLRs in AGNs at $z \sim$ 3 is solar or
super-solar, without assuming the gas density and ionization parameter thanks
to the constrains from the faint emission lines. This result suggests that
high-$z$ radio galaxies are already chemically matured at $z \sim$ 3.
