A Differential Abundance Analysis of Very Metal-poor Stars* **

We have performed a differential, line-by-line, chemical abundance analysis, ultimately relative to the Sun, of nine very metal-poor main sequence halo stars, near [Fe/H]=$-$2 dex. Our abundances range from $-2.66\leq\mathrm{[Fe/H]}\leq-1.40$ dex with conservative uncertainties of 0.07 dex. We find an average [$\alpha$/Fe]$=0.34\pm0.09$ dex, typical of the Milky Way. While our spectroscopic atmosphere parameters provide good agreement with HST parallaxes, there is significant disagreement with temperature and gravity parameters indicated by observed colors and theoretical isochrones. Although a systematic underestimate of the stellar temperature by a few hundred degrees could explain this difference, it is not supported by current effective temperature studies and would create large uncertainties in the abundance determinations. Both 1D and $\langle$3D$\rangle$ hydrodynamical models combined with separate 1D non-LTE effects do not yet account for the atmospheres of real metal-poor MS stars, but a fully 3D non-LTE treatment may be able to explain the ionization imbalance found in this work.