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 d
ue 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.
We study the link between the kinematic-morphology of galaxies, as inferred from integral-field stellar kinematics, and their relation between mass and star formation rate (SFR). Our sample consists of $sim 3200$ galaxies with integral-field spectros
copic data from the MaNGA survey with available determinations of their effective stellar angular momentum within the half-light radius $lambda_{R_e}$. We find that for star-forming galaxies, namely along the star formation main sequence (SFMS), the $lambda_{R_e}$ values remain large and almost unchanged over about two orders of magnitude in stellar mass, with the exception of the lowest masses $mathcal{M}_{star}lesssim2times10^{9} mathcal{M}_{odot}$, where $lambda_{R_e}$ slightly decreases. The SFMS is dominated by spiral galaxies with small bulges. Below the SFMS, but above the characteristic stellar mass $mathcal{M}_{rm crit}approx2times10^{11} mathcal{M}_{odot}$, there is a sharp decrease in $lambda_{R_e}$ with decreasing star formation rate: massive galaxies well below the SFMS are mainly slow-rotator early-type galaxies, namely genuinely spheroidal galaxies without disks. Below the SFMS and below $mathcal{M}_{rm crit}$ the decrease of $lambda_{R_e}$ with decreasing SFR becomes modest or nearly absent: low-mass galaxies well below the SFMS, are fast-rotator early-type galaxies, and contain fast-rotating stellar disks like their star-forming counterparts. We also find a small but clear environmental dependence for the massive galaxies: in the mass range $10^{10.9}-10^{11.5} mathcal{M}_{odot}$, galaxies in rich groups or denser regions or classified as central galaxies have lower values of $lambda_{R_e}$. While no environmental dependence is found for galaxies of lower mass. We discuss how our results can be understood as due to the different star formation and mass assembly histories of galaxies with varying mass.
