Relation between the dynamics and the flattening of elliptical galaxies

The interrelations among the eccentricity of equidensity surfaces, the anisotropy of mean square velocities, and the mass distribution function f (assumed to be a function only of energy E and angular momentum J about the axis of symmetry) are studied. Distribution functions that are separable in E and J, such as are often assumed, give rise to eccentricity profiles in which the eccentricity declines substantially in the central regions. Conversely, if the eccentricity is to remain relatively uniform in the central regions, then the dependence of f on J must grow in relative significance at high energies. Properties of finite truncated isothermal spheres are also investigated. When their distribution functions are truncated smoothly in phase space, their radii, in particular, are shown to be curiously sensitive to the central potential, and able to vary by more than a factor of 10.