Parallax of a Mira variable R Ursae Majoris studied with astrometric VLBI

We have measured an annual parallax of the Mira variable R~Ursae~Majoris (R~UMa) with the VLBI exploration for Radio Astronomy (VERA). From the monitoring VLBI observations spanning about two years, we detected H$_2$O maser spots in the LSR velocities ranges from 37 to 42 km\,s$^{-1}$. We derived an annual parallax of 1.97$\pm$0.05\,mas, and it gives a corresponding distance of 508$\pm$13\,pc. The VLBI maps revealed 72 maser spots distributed in $\sim$110 au area around an expected stellar position. Circumstellar kinematics of the maser spots were also revealed by subtracting a systemic motion in the Hipparcos catalog from proper motions of each maser spots derived from our VLBI observations. Infrared photometry is also conducted to measure a $K$ band apparent magnitude, and we obtained a mean magnitude of $m_K$ = 1.19$\pm$0.02\,mag. Using the trigonometric distance, the $m_K$ is converted to a $K$ band absolute magnitude of $M_K = -$7.34$\pm$0.06\,mag. This result gives a much more accurate absolute magnitude of R~UMa than previously provided. We solved a zero-point of $M_K - \log P$ relation for the Galactic Mira variables and obtained a relation of $M_K = -$3.52 $\log P$ + (1.09 $\pm$ 0.14). Other long period variables including red supergiants, whose distances were determined from astrometric VLBI, were also compiled to explore the different sequences of $M_K - \log P$ relation.