Metallicity Distribution in M31 and M33 Based on the Tip-RGB Near-infrared Color

This study investigates the metallicity distribution in M31 and M33 by using the near-infrared color index $J-K$ of tip-red-giant-branch (TRGB) of the member stars from \cite{2021ApJ...907...18R} after removing the foreground dwarf stars by the near-infrared $J-H/H-K$ diagram as well as the Gaia astrometric measurements. We employ the Voronoi binning technique to divide the galaxy into sub-regions, the PN method to determine the TRGB position in the $J-K$/$K$ diagram, and the bootstrap method to estimate the uncertainties. The TRGB positions are calculated to be $J-K = 1.195 \pm 0.002$ and $1.100 \pm 0.003$, and $K = 17.615 \pm 0.007$ and $18.185 \pm 0.053$ for M31 and M33 respectively as an entity, consistent with previous results. The $J-K$ gradient of M31 is found to be $-$0.0055 kpc$^{-1}$ at $R_{\rm GC}=(0, 24)$ kpc and $-$0.0002 kpc$^{-1}$ at $R_{\rm GC}=(24, 150)$ kpc. Notably, two dust rings are evident at 11.5 kpc and 14.5 kpc, in agreement with previous studies. The $J-K$ of M33 is analyzed in four directions and generally shows a trend of gradually decreasing from the inside to the outside, however, it shows an increasing trend of 0.022 kpc$^{-1}$ in the inner 0$-$2 kpc in the west. Through the color$-$metallicity relation of TRGB, the metallicity gradient of M31 turns to be $-0.040 \pm 0.0012$ dex kpc$^{-1}$ with $R_{\rm GC}<30$ kpc and $-0.001 \pm 0.0002$ dex kpc$^{-1}$ with $R_{\rm GC}>30$ kpc, and for M33, $-$0.269 $\pm$ 0.0206 dex kpc$^{-1}$ with $R_{\rm GC}<9$ kpc.