The phase-space distribution function for weakly flattened axisymmetric stellar systems

The astrophysical importance of galaxy modelling lies in the role it plays in understanding the structure, formation, and evolution of galaxies. Models can be constructed numerically or analytically (e.g. through the Jeans equations). Some analytical approach is based on the use of the phase-space Distribution Function (DF), whose positivity is a necessary condition for the physical consistency of a galaxy model. This criterion allows physically unacceptable models to be discarded even when they provide a good fit to the observational data. Moreover, an analytical DF enables the computation of velocity moments of all orders, providing detailed predictions for stellar kinematics. According to the Jeans theorem, the DF depends on the phase-space coordinates through the integrals of motion in the total gravitational potential. However, the relation between the DF and observable quantities is expressed by complicated integral equations, making its determination an inversion problem. To date, the only case for which a general inversion formula is available is spherical symmetry, through the Eddington formula. In axial symmetry, existing methods rely on analytical continuation into the complex plane, limiting their applicability to observational data. In this thesis, we develop a novel inversion method for two-integral axisymmetric stellar systems by considering small departures from spherical symmetry. In this regime, density and gravitational potential admit homoeoidal expansions, where the flattening parameter appears explicitly and the geometry separates naturally into spherical and non-spherical contributions. By exploiting this structure, we show that the inversion equations can be decomposed into terms that closely resemble the Eddington formula. This leads to a new inversion formula for the DF without introducing complex variables, providing a promising framework for the analytical modelling of axisymmetric galaxies.