ALMA Astrometry of the Objects within 0.5 pc of Sagittarius A^ast

Atacama Large Millimeter/Submillimeter Array (ALMA) is promising to be a powerful tool for precision astrometry of the area around Sagittarius A$^\ast$ (Sgr A$^\ast$) because it has the high angular resolution, high sensitivity, and wide field of view. We have observed the area including the Nuclear Star Cluster at 230 GHz with ALMA in October 2017. The angular resolution is ~0.03". We determined the relative positions to Sgr A$^\ast$ of 65 compact objects in the area with the accuracy of 0.001". We also analyzed the similar ALMA archival data obtained in June 2019 and determined the 64 relative positions in these objects. We derived the proper motions relative to Sgr A$^\ast$ by comparing these positions. The derived proper motions are roughly described with both clockwise and counterclockwise rotations around Sgr A$^\ast$. The rotation velocities are reproduced by Kepler orbits bounded around Sgr A$^\ast$. Moreover, the proper motions include co-moving clusters for example IRS13E and IRS13N. The positions and proper motions are almost consistent with those by previous infrared observations. Therefore the observational demonstrations would prove that ALMA is a powerful tool for precision astrometry of the region.