Implication of the period-magnitude relation for massive AGB stars and its astronomical applications

We present astrometric very long baseline interferometry (VLBI) studies of AGB stars. To understand the properties and evolution of AGB stars, distances are an important parameter. The distribution and kinematics of their circumstellar matter are also revealed with the VLBI method. We used the VERA array to observe 22\,GHz H$_2$O masers in various subclasses of AGB stars. Parallaxes of the three OH/IR stars NSV17351, OH39.7$+$1.5, IRC$-$30363, and the Mira-type variable star AW~Tau were newly obtained. We present the circumstellar distribution and kinematics of H$_2$O masers around NSV17351. The absolute magnitudes in mid-infrared bands of OH/IR stars with very long pulsation periods were investigated and a period-magnitude relation in the WISE W3 band, $M_{\mathrm{W3}} = (-7.21\pm1.18)\log P + (9.25\pm3.09)$, was found for the Galactic AGB stars. The VLBI is still a powerful tool for parallax measurements of the Galactic AGB stars surrounded by thick dust shells.