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In the $(\lambda_{\rm R}, \varepsilon)$ and $(V/\sigma, \varepsilon)$diagrams for characterizing dynamical states, the fast-rotator galaxies (bothearly-type and spirals) are distributed within a well-defined leaf-shapedenvelope. This was explained as due to an upper limit to the orbital anisotropyincreasing with galaxy intrinsic flattening. However, a physical explanationfor this empirical trend was missing. Here we construct Jeans AnisotropicModels (JAM), with either cylindrically or spherically aligned velocityellipsoid (two extreme assumptions), and each with either spatially-constant or-variable anisotropy. We use JAM to build mock samples of axisymmetricgalaxies, assuming on average an oblate shape for the velocity ellipsoid (asrequired to reproduce the rotation of real galaxies), and limiting the radialanisotropy $\beta$ to the range allowed by physical solutions. We find that allfour 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 quitedifferent models, we conclude that the empirical anisotropy upper limit in realgalaxies, and the corresponding observed distributions in the $(\lambda_{\rmR}, \varepsilon)$ and $(V/\sigma, \varepsilon)$ diagrams, are due to the lackof physical axisymmetric equilibrium solutions at high $\beta$ anisotropy whenthe velocity ellipsoid is close to oblate.

monthly notices of the royal astronomical society. letters

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