Jeans and Boltzmann Solutions for Oblate Galaxies with Flat Rotation Curves

Abstract

A general solution of the Jeans equations for oblate scale-free logarithmic potentials is given. This provides all possible second velocity moments that can hold up a stellar population of flattened scale-free density against the gravity field. A two-parameter subset of second moments for the self-consistent density of Binney's model is examined in detail. These solutions have the desirable property that the observable dispersions in the radial and proper motions can be given explicitly. In the spherical limit, the potential of these models reduces to that of the singular isothermal sphere. The Jeans solutions for scale-free densities of arbitrary flattening that can correspond to physical three-integral distribution functions are identified. The problem of finding distribution functions associated with the Jeans solutions in flattened scale-free logarithmic potentials is then investigated for Binney's model. An approximate solution of the collisionless Boltzmann equation is found which provides a third (partial) integral of good accuracy for thin and near-thin tube orbits. It is a modification of the total angular momentum. This enables the construction of many simple three-integral distribution functions. The kinematic properties of these approximate DFs are shown to agree with a subset of the Jeans solutions --- which are thereby confirmed as good approximations to physical solutions.