Jeans modelling of weakly flattened ellipsoidal systems

In the homoeoidal expansion, a given ellipsoidally stratified density distribution and its associated potential are expanded in the (small) density flattening parameter eta, and usually truncated at the linear order. The truncated density-potential pair obeys exactly the Poisson equation, and it can be interpreted as the first-order expansion of the original ellipsoidal density-potential pair, or as a new autonomous system. In the first interpretation, in the solutions of the Jeans equations the quadratic terms in eta must be discarded ('eta-linear' solutions), while in the second ('eta-quadratic') all terms are retained. In this work, we study the importance of the quadratic terms by using the ellipsoidal Plummer model and the Perfect Ellipsoid, which allow for fully analytical eta-quadratic solutions. These solutions are then compared with those obtained numerically for the original ellipsoidal models, finding that the eta-linear models already provide an excellent approximation of the numerical solutions. As an application, the eta-linear Plummer model (with a central black hole) is used for the phenomenological interpretation of the dynamics of the weakly flattened and rotating globular cluster NGC 4372, confirming that this system cannot be interpreted as an isotropic rotator, a conclusion reached previously with more sophisticated studies.