Probing M subdwarf metallicity with an esdK5+esdM5.5 binary

We present a spectral analysis of the binary G 224-58 AB that consists of the coolest M extreme subdwarf (esdM5.5) and a brighter primary (esdK5). This binary may serve as a benchmark for metallicity measurement calibrations and as a test-bed for atmospheric and evolutionary models for esdM objects. We determine abundances primarily using high resolution optical spectra of the primary. Other parameters were determined from the fits of synthetic spectra computed with these abundances to the observed spectra from 0.4 to 2.5 microns for both components. We determine \Tef =4625 $\pm$ 100 K, \logg = 4.5 $\pm$ 0.5 for the A component and \Tef = 3200 $\pm$ 100 K, \logg = 5.0 $\pm$ 0.5, for the B component. We obtained abundances of [Mg/H]=$-$1.51$\pm$0.08, [Ca/H]=$-$1.39$\pm$0.03, [Ti/H]=$-$1.37$\pm$0.03 for alpha group elements and [CrH]=$-$1.88$\pm$0.07, [Mn/H]=$-$1.96$\pm$0.06, [Fe/H]=$-$1.92$\pm$0.02, [Ni/H]=$-$1.81$\pm$0.05 and [Ba/H]W=$-$1.87$\pm$0.11 for iron group elements from fits to the spectral lines observed in the optical and infrared spectral regions of the primary. We find consistent abundances with fits to the secondary albeit at lower signal-to-noise. Abundances of elements in \ga and \gb atmospheres cannot be described by one metallicity parameter. The offset of $\sim$ 0.4 dex between the abundances derived from alpha element and iron group elements corresponds with our expectation for metal-deficient stars. We thus clarify that some indices used to date to measure metallicities for establishing esdM stars based on CaH, MgH and TiO band system strength ratios in the optical and H$_2$O in the infrared relate to abundances of alpha-element group rather than to iron peak elements. For metal deficient M dwarfs with [Fe/H]