Metallicity Measurements of Gamma-Ray Burst and Supernova Explosion
  Sites: Lessons from HII regions in M31
  We examine how the small-scale ($<$\,kpc) variation of metallicity within a
galaxy, which is found in nearby galaxies, affect the observational estimates
of metallicity in the explosion sites of transient events such as core-collapse
supernovae (CC SNe) and gamma-ray bursts (GRBs). Assuming the same luminosity,
metallicity, and spatial distributions of \ion{H}{ii}\ regions (hereafter
HIIRs) as observed in M31, we compute the apparent metallicities that we would
obtain when the spectrum of a target region is blended with those of
surrounding HIIRs within the length scale of typical spatial resolution. When
the spatial resolution of spectroscopy is $\lesssim$ 0.5\,kpc, which is typical
for the existing studies of CC SN sites, we find that the apparent
metallicities reflect the metallicities of target regions, but with significant
systematic uncertainties in some cases. When the spatial resolution is
$\gtrsim$ 1.0 kpc, regardless of the target regions (which has a wide range of
metallicity that spans $\sim$ 0.6 dex for the M31 HIIRs), we always obtain the
apparent metallicities similar to the average metallicity of the M31 HIIRs.
Given that the apparent metallicities measured with $\gtrsim$\,kpc scale
resolution do not necessarily reflect the immediate environment of the stellar
explosions, the current observational estimates of high metallicities for some
of the long GRB host galaxies do {\em not} rule out the hypothesis that the
long GRBs are exclusively born in a low-metallicity environment.
