SCALE-FREE DYNAMICAL MODELS FOR GALAXIES - FLATTENED DENSITIES IN SPHERICAL POTENTIALS

This paper presents two families of phase-space distribution functions that generate scale-free spheroidal mass densities in scale-free sphe rical potentials. The assumption of a spherical potential has the adva ntage that all integrals of motion are known explicitly. The 'case I' distribution functions are anisotropic generalizations of the flattene d f(E,L(z)) model, which they include as a special case. The 'case II' distribution functions generate flattened constant-anisotropy models. Free parameters control the radial power-law slopes of the mass densi ty and potential, the flattening of the mass distribution, and the vel ocity dispersion anisotropy. The models can describe the outer parts o f galaxies and the density cusp structure near a central black hole, b ut also provide general insight into the dynamical properties of flatt ened systems. Because of their simplicity they provide a useful comple mentary approach to the construction of flattened self-consistent thre e-integral models for elliptical galaxies. The dependence of the intri nsic and projected properties on the model parameters and the inclinat ion is described. The case I models have a larger ratio of rms tangent ial to radial motion in the equatorial plane than on the symmetry axis , the more so for smaller axial ratios. The case II models have a cons tant ratio of rms tangential to radial motion throughout the system, a s characterized by Binney's parameter beta. The maximum possible ratio upsilon(p)/sigma(p) Of the mean projected line-of-sight velocity and velocity dispersion on the projected major axis always decreases with increasing radial anisotropy. The observed ratio of the rms projected line-of-sight velocities on the projected major and minor axes of elli ptical galaxies is best fitted by the case II models with beta greater than or similar to 0. These models also predict non-Gaussian velocity profile shapes consistent with existing observations. The distributio n functions are used to model the galaxies NGC 2434 (E1) and NGC 3706 (E4), for which stellar kinematical measurements out to two effective radii indicate the presence of dark haloes. The velocity profile shape s of both galaxies can be well fitted by radially anisotropic case II models with a spherical logarithmic potential. This contrasts with the f(E,L(z)) models studied previously, which require flattened dark hal oes to fit the data.