The Detection of [O III]$\lambda$4363 in a Lensed, Dwarf Galaxy at $z$ = 2.59: Testing Metallicity Indicators and Scaling Relations at High Redshift and Low Mass

We present Keck/MOSFIRE and Keck/LRIS spectroscopy of A1689-217, a lensed (magnification $\sim7.9$), star-forming (SFR $\sim$ 16 $\rm M_\odot\ yr^{-1}$), dwarf (log($M_\ast/M_\odot$) = $8.07-8.59$), Ly$\alpha$-emitter ($\rm EW_0\sim138$ A) at $z$ = 2.5918. Dwarf galaxies similar to A1689-217 are common at high redshift and likely responsible for reionization, yet few have been studied with detailed spectroscopy. We report a 4.2$\sigma$ detection of the electron-temperature-sensitive [O III]$\lambda$4363 emission line and use this line to directly measure an oxygen abundance of 12+$\log$(O/H) = 8.06 $\pm$ 0.12 ($\sim1/4\ \rm Z_\odot$). A1689-217 is the lowest mass galaxy at $z>$ 2 with an [O III]$\lambda$4363 detection. Using the rest-optical emission lines, we measure A1689-217's other nebular conditions including electron temperature ($T_e$([O III]) $\sim$ 14,000 K), electron density ($n_e\sim$ 220 $\rm cm^{-3}$) and reddening ($E(B-V)\sim$ 0.39). We study relations between strong-line ratios and direct metallicities with A1689-217 and other galaxies with [O III]$\lambda$4363 detections at $z\sim0-3.1$, showing that the locally-calibrated, oxygen-based, strong-line relations are consistent from $z\sim0-3.1$. We also show additional evidence that the $\rm O_{32}$ vs. $\rm R_{23}$ excitation diagram can be utilized as a redshift-invariant, direct-metallicity-based, oxygen abundance diagnostic out to $z\sim$ 3.1. From this excitation diagram and the strong-line ratio $-$ metallicity plots, we observe that the ionization parameter at fixed O/H is consistent with no redshift evolution. Although A1689-217 is metal-rich for its $M_\ast$ and SFR, we find it to be consistent within the large scatter of the low-mass end of the Fundamental Metallicity Relation.