  Red supergiants (RSGs) are evolved massive stars that represent extremes, in
both their physical sizes and their cool temperatures, of the massive star
population. Effective temperature ($T_{\rm eff}$) is the most critical physical
property needed to place a RSG on the Hertzsprung-Russell Diagram, due to the
stars' cool temperatures and resulting large bolometric corrections. Several
recent papers have examined the potential utility of atomic line equivalent
widths in cool supergiant spectra for determining $T_{\rm eff}$ and other
physical properties (dorda2016b, dorda2016a) and found strong correlations
between Ti I and Fe I spectral features and $T_{\rm eff}$ in earlier-type cool
supergiants (G and early K) but poor correlations in M-type stars, a spectral
subtype that makes up a significant fraction of RSGs. We have extended this
work by measuring the equivalent widths of Ti, Fe, and Ca lines in late K- and
M-type RSGs in the Milky Way, Large Magellanic Cloud, and Small Magellanic
Cloud, and compared these results to the predictions of the MARCS stellar
atmosphere models. Our analyses show a poor correlation between $T_{\rm eff}$
and the Fe I and Ti I lines in our observations (at odds with strong
correlations predicted by stellar atmosphere models), but do find statistically
significant correlations between $T_{\rm eff}$ and the Ca II triplet (CaT)
features of Milky Way RSGs, suggesting that this could be a potential
diagnostic tool for determining $T_{\rm eff}$ in M type supergiants. We also
examine correlations between these spectral features and other physical
properties of RSGs (including metallicity, surface gravity, and bolometric
magnitude), and consider the underlying physics driving the evolution of atomic
line spectra in RSGs.
