oMEGACat. VI. Analysis of the overall kinematics of Omega Centauri in 3D: velocity dispersion, kinematic distance, anisotropy, and energy equipartition

Omega Centauri ($\omega$ Cen) is the Milky Way's most massive globular cluster and is likely the stripped nucleus of an accreted dwarf galaxy. In this paper, we analyze $\omega$ Cen's kinematics using data from oMEGACat, a comprehensive catalog of $\omega$ Cen's central regions, including 1.4 million proper motion measurements and 300,000 spectroscopic radial velocities. Our velocity dispersion profiles and kinematic maps are consistent with previous work but improve on their resolution, precision, and spatial coverage. The cluster's 3D dispersion is isotropic in the core, with increasing radial anisotropy at larger radii. The 2D kinematic maps show an elongation of the velocity dispersion field comparable to the flattening observed photometrically. We find good agreement between proper motions and line-of-sight velocity dispersion and measure a kinematic distance of 5494$\pm$61 pc, the most precise kinematic distance to $\omega$ Cen available. The subset of data with precise metallicity measurements shows no correlation between metallicity and kinematics, supporting the picture of well-mixed stellar populations within the half-light radius of $\omega$ Cen. Finally, we study the degree of energy equipartition using a large range of stellar masses. We find partial energy equipartition in the center that decreases towards large radii. The spatial dependence of the radial energy equipartition is stronger than the tangential energy equipartition. Our kinematic observations can serve as a new reference for future dynamical modeling efforts that will help to further disentangle the complex mass distribution within $\omega$ Cen.