Mid-infrared Period--Luminosity relations of Gaia DR3 Long Period Variables

Long period variable stars (LPVs) are very promising distance indicators in the infrared bands. We selected asymptotic giant branch (AGB) stars in the Large and Small Magellanic Cloud (LMC and SMC) from the Gaia Data Release 3 (DR3) LPV catalog, and classified them into oxygen-rich (O-rich) and carbon-rich (C-rich) AGB stars. Using the WISE database, we determined the $W1$- and $W2$-bands Period--Luminosity relations (PLRs) for each pulsation-mode sequence of AGB stars. The dispersion of the PLRs of O-rich AGB stars in sequences $\rm{C^{'}}$ and C is relatively small, around 0.14 mag. The PLRs of LMC and SMC are consistent in each sequence. % while due to selection effect, {\bf the PLR zero points} of Milky Way are significantly underestimated. In the $W2$ band, the PLR of large-amplitude C-rich AGB stars is steeper than that of small-amplitude C-rich AGB stars, due to their more circumstellar dust. By two methods, we find that some PLR sequences of O-rich AGB stars in the LMC are dependent on metallicity. The coefficients of the metallicity effect are $\beta = -0.533 \pm 0.213 $ mag/dex and $\beta= -0.767 \pm 0.158$ mag/dex for sequence C in $W1$ and $W2$ bands, respectively. The significance of the metallicity effect in $W1$ band for the four sequences is $2.2-3.5 \sigma$. Both of these imply that distance measurements using O-rich Mira maybe need to take the metallicity effect into account.