Physical Explanation for the Galaxy Distribution on the $(\lambda_{\rm R}, \varepsilon)$ and $(V/\sigma, \varepsilon)$ Diagrams or for the Limit on Orbital Anisotropy

In the $(\lambda_{\rm R}, \varepsilon)$ and $(V/\sigma, \varepsilon)$ diagrams for characterizing dynamical states, the fast-rotator galaxies (both early-type and spirals) are distributed within a well-defined leaf-shaped envelope. This was explained as due to an upper limit to the orbital anisotropy increasing with galaxy intrinsic flattening. However, a physical explanation for this empirical trend was missing. Here we construct Jeans Anisotropic Models (JAM), with either cylindrically or spherically aligned velocity ellipsoid (two extreme assumptions), and each with either spatially-constant or -variable anisotropy. We use JAM to build mock samples of axisymmetric galaxies, assuming on average an oblate shape for the velocity ellipsoid (as required to reproduce the rotation of real galaxies), and limiting the radial anisotropy $\beta$ to the range allowed by physical solutions. We find that all four mock samples naturally predict the observed galaxy distribution on the $(\lambda_{\rm R}, \varepsilon)$ and $(V/\sigma, \varepsilon)$ diagrams, without further assumptions. Given the similarity of the results from quite different models, we conclude that the empirical anisotropy upper limit in real galaxies, and the corresponding observed distributions in the $(\lambda_{\rm R}, \varepsilon)$ and $(V/\sigma, \varepsilon)$ diagrams, are due to the lack of physical axisymmetric equilibrium solutions at high $\beta$ anisotropy when the velocity ellipsoid is close to oblate.