Examining the relationships between colour, $T_{\rm eff}$, and [M/H] for APOGEE K and M dwarfs

We present the effective temperatures ($T_{\rm eff}$), metallicities, and colours in SDSS, 2MASS, and WISE filters, of a sample of 3834 late-K and early-M dwarfs selected from the Sloan Digital Sky Survey APOGEE spectroscopic survey ASPCAP catalog. We confirm that ASPCAP $T_{\rm eff}$ values between 3550 K$<T_{\rm eff}<$4200 K are accurate to $\sim$100 K compared to interferometric $T_{\rm eff}$ values. In that same $T_{\rm eff}$ range, ASPCAP metallicities are accurate to 0.18 dex between $-1.0<$[M/H]$<0.2$. For these cool dwarfs, nearly every colour is sensitive to both $T_{\rm eff}$ and metallicity. Notably, we find that $g-r$ is not a good indicator of metallicity for near-solar metallicity early-M dwarfs. We confirm that $J-K_S$ colour is strongly dependent on metallicity, and find that $W1-W2$ colour is a promising metallicity indicator. Comparison of the late-K and early-M dwarf colours, metallicities, and $T_{\rm eff}$ to those from three different model grids shows reasonable agreement in $r-z$ and $J-K_S$ colours, but poor agreement in $u-g$, $g-r$, and $W1-W2$. Comparison of the metallicities of the KM dwarf sample to those from previous colour-metallicity relations reveals a lack of consensus in photometric metallicity indicators for late-K and early-M dwarfs. We also present empirical relations for $T_{\rm eff}$ as a function of $r-z$ colour combined with either [M/H] or $W1-W2$ colour, and for [M/H] as a function of $r-z$ and $W1-W2$ colour. These relations yield $T_{\rm eff}$ to $\sim$100 K and [M/H] to $\sim$0.18 dex precision with colours alone, for $T_{\rm eff}$ in the range of 3550-4200 K and [M/H] in the range of $-$0.5-0.2.