The Construction of Non-Spherical Models of Quasi-Relaxed Stellar Systems

Spherical models of collisionless but quasi-relaxed stellar systems have long been studied as a natural framework for the description of globular clusters. Here we consider the construction of self-consistent models under the same physical conditions, but including explicitly the ingredients that lead to departures from spherical symmetry. We take a stellar system on a circular orbit inside a galaxy represented as a ``frozen" external tidal field. The equilibrium distribution function is obtained from the one describing the spherical case by replacing the energy integral with the relevant Jacobi integral. The construction of the model requires the investigation of a singular perturbation problem for an elliptic partial differential equation with a free boundary, for which we provide an explicit solution to two orders. We outline the relevant parameter space, thus opening the way to a systematic study of the properties of a two-parameter family of physically justified non spherical models of quasi-relaxed stellar systems. The general method developed here can also be used to construct models for which the non-spherical shape is due to internal rotation. Eventually, the models will be a useful tool to investigate whether the shapes of globular clusters are primarily determined by internal rotation, by external tides, or by pressure anisotropy.