100K and 0.5 dex in temperatures and gravity $+$ metallicity, respectively. We ignored regions of the observed spectra strongly affected by telluric bands, in particular the 530–570 nm, 928–1010 nm, 1110–1150 nm, 1340–1460 nm, 1790–1970 nm wavelength ranges.

We shifted our observed spectra to the wavelength of the models. We calculated the heliocentric radial velocities for our sample of metal-poor M dwarfs using a set of about 15 lines (potassium, sodium, iron, and calcium) in the 610–840 nm wavelength range, the exact number depending on the quality of the spectrum and strength of the line (Table \[tab\_MplusTeff\_sdM:table\_param\]). We measured consistent shifts between these five strong lines for all sources, with dispersions of the order of a few km/s. We assume that the true error bars are set by the resolution of our X-shooter spectra (3900–6700), corresponding to radial velocity uncertainties of approximately 15–25 km/s.

Then, we minimized the $\chi^2$ value for each observed and theoretical spectrum, as:

$$\label{eq:chi2}
\chi^2 = \frac{1}{N} \sum_i\left\{ \frac{ ({\rm F}_{\rm obs,i} - A \ {\rm F}_{\rm mod,i})^2}{(\Delta{\rm F}_{\rm obs,i})^2}\right\}$$

The scale factor $A$ is calculated to minimize $\chi^2$ for each case as:

$$\label{eq:chi2b}
A = \frac{ \sum_i({\rm F}_{\rm obs,i} {\rm F}_{\rm mod,i}/ \Delta{\rm F}_{\rm obs,i}^2)}{\sum_i({\rm 