Giant stars are the most common contaminats of colour-based selections of M dwarfs. A way to identify giants, when their distances are unknown, is by their location in a reduced proper motion diagram \citep[e.g.][]{2011AJ....142..138L}. To get estimates of the proper motions of the 12<\(K_s<\)16 M dwarf colour-based selection, we performed a search within 5\(\arcsec\) of their coordinates in the PPMXL Catalog \citep{2010AJ....139.2440R}. We obtained total proper motions, \(\mu\) in \(\arcsec\)/yr, for 8,728 out of the 9,232 stars of the 12<\(K_s\)<16 M dwarf selection. We calculated their J magnitude reduced proper motion \(H_J\) using the definition
\(H_J\) = \(J\) + 5 log10()
\citet{2011AJ....142..138L} defined a criterion to separate M dwarfs from giants based on \(V\) magnitude reduced proper motion \(H_V\) and \((V-J)\) colour. This criterion cannot be used for our stars since we do not count with \(V\) magnitudes for the 9,232 colour-selected M dwarfs. For that reason, we computed an equivalent criterion based on \(J\) and \(K_s\) magnitudes. We grouped the stars of the \citet{2011AJ....142..138L} study by their estimated spectral types, obtained their mean \((V-J)\) and \((J-K_s)\) colours, and calculated the dwarf/giant discriminator \(H_J^{*}\) as function of mean \((V-J)\) per spectral type, using our definition of \(H_J\), by re-writting Equation 8 of \citet{2011AJ....142..138L} as following
\(H_J^{*}\) = 1.5 \((V-J)\) - 3.0
Then, using the mean \((J-K_s)\) colour corresponding to each mean \((V-J)\) per spectral type, we performed a linear fit to get \(H_J^{*}\) as function of \((J-K_s)\), and, therefore, an equivalent criterion to equation XX based on \((J-K_s)\), instead of \((V-J)\)
\(H_J^{dwarf}\) > \(H_J^{*}\) = 68.5 \((J-K_s)\) - 50.7
Using the criterion above, we identified 771 likely giant stars in the 12<\(K_s\)<16 colour-selected M dwarf sample. Their locations in the CMD and \((H-K_s, J-H)\) diagram are shown in Figure XX.