for lower metallicities. For direct comparison, we limited the wavelength range to 450–2500 nm and smoothed the synthetic SEDs with the Interactive Data Language (IDL) function[^6] to the spectral resolution of our X-shooter spectra.

Throughout the paper, we use the term metallicity \[Fe/H\] which is given by the BT-Settl models. We do not infer abundances or metallicities of single element but the global values given by the synthetic spectra. The former can differ from the latter as shown for a metal-poor low-mass binary [@pavlenko15]. The study of abundances of individual elements in subdwarfs is beyond the scope of this paper but will be investigated in a future publication.

Comparison: observations vs. models {#MplusTeff_sdM:Comparison}
===================================

We fitted the observed spectra with the BT-Settl SEDs with a chi-square ($\chi^2$) minimisation procedure described in Sect. \[MplusTeff\_sdM:Comparison\_chi2\]. We derived the temperature (Sect. \[MplusTeff\_sdM:Comparison\_Teff\]) and metallicity (Sect. \[MplusTeff\_sdM:Comparison\_MH\]) scales from the best model fits.

Chi-square fitting {#MplusTeff_sdM:Comparison_chi2}
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We performed a $\chi^2$ fit to compare the observed spectra with those in the BT-Settl theoretical library. We considered the following ranges in temperature, gravity, and metallicity: 4000–2500K, $\log$g=4.5–6.0 dex, and \[M/H\] between $-$3.0 and 0.0 dex, as expected for old low-mass M-type dwarfs. The steps are 