The ellipsoid of stellar random motions is a fundamental ingredient of galaxy dynamics. Yet it has long been difficult to constrain this component in disks others than the Milky Way. This article presents the modeling of the azimuthal-to-radial axis ratio of the velocity ellipsoid of galactic disks from stellar dispersion maps using integral field spectroscopy data of the CALIFA survey. The measured azimuthal anisotropy is shown to be not strongly dependent on the assumed vertical-to-radial dispersion ratio of the ellipsoid. The anisotropy distribution shows a large diversity in the orbital structure of disk galaxies from tangential to radial stellar orbits. Globally, the orbits are isotropic in inner disk regions and become more radial as a function of radius, although this picture tends to depend on galaxy morphology and luminosity. The Milky Way orbital anisotropy profile measured from the Second Gaia Data Release is consistent with those of CALIFA galaxies. A new correlation is evidenced, linking the absolute magnitude or stellar mass of the disks to the azimuthal anisotropy. More luminous disks have more radial orbits and less luminous disks have isotropic and somewhat tangential orbits. This correlation is consistent with the picture in galaxy evolution in which orbits become more radial as the mass grows and is redistributed as a function of time. With the help of circular velocity curves, it is also shown that the epicycle theory fails to reproduce the diversity of the azimuthal anisotropy of stellar random motions, as it predicts only nearly radial orbits in the presence of flat curves. The origin of this conflict is yet to be identified. It also questions the validity of the vertical-to-radial axis ratio of the velocity ellipsoid derived by many studies in the framework of the epicyclic approximation.
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