The construction of non-spherical models of quasi-relaxed stellar systems

摘要

Spherical models of collisionless but quasi-relaxed stellar systems have longbeen studied as a natural framework for the description of globular clusters.Here we consider the construction of self-consistent models under the samephysical conditions, but including explicitly the ingredients that lead todepartures from spherical symmetry. In particular, we focus on the effects ofthe tidal field associated with the hosting galaxy. We then take a stellarsystem on a circular orbit inside a galaxy represented as a "frozen" externalfield. The equilibrium distribution function is obtained from the onedescribing the spherical case by replacing the energy integral with therelevant Jacobi integral in the presence of the external tidal field. Then theconstruction of the model requires the investigation of a singular perturbationproblem for an elliptic partial differential equation with a free boundary, forwhich we provide a method of solution to any desired order, with explicitsolutions to two orders. We outline the relevant parameter space, thus openingthe way to a systematic study of the properties of a two-parameter family ofphysically justified non-spherical models of quasi-relaxed stellar systems. Thegeneral method developed here can also be used to construct models for whichthe non-spherical shape is due to internal rotation. Eventually, the modelswill be a useful tool to investigate whether the shapes of globular clustersare primarily determined by internal rotation, by external tides, or bypressure anisotropy.