Self-consistent flattened isochrones

Abstract

We present a family of self-consistent axisymmetric stellar systems that have analytic distribution functions (DFs) of the form f(J), so they depend on three integrals of motion and have triaxial velocity ellipsoids. The models, which are generalizations of Hénon's isochrone sphere, have four dimensionless parameters, two determining the part of the DF that is even in Lz and two determining the odd part of the DF (which determines the azimuthal velocity distribution). Outside their cores, the velocity ellipsoids of all models tend to point to the model's centre, and we argue that this behaviour is generic, so near the symmetry axis of a flattened model, the long axis of the velocity ellipsoid is naturally aligned with the symmetry axis and not perpendicular to it as in many published dynamical models of well-studied galaxies. By varying one of the DF parameters, the intensity of rotation can be increased from zero up to a maximum value set by the requirement that the DF be non-negative. Since angle–action coordinates are easily computed for these models, they are ideally suited for perturbative treatments and stability analysis. They can also be used to choose initial conditions for an N-body model that starts in perfect equilibrium, and to model observations of early-type galaxies. The modelling technique introduced here is readily extended to different radial density profiles, more complex kinematics and multicomponent systems. A number of important technical issues surrounding the determination of the models’ observable properties are explained in two appendices.