Three-integral models of oblate elliptical galaxies

Abstract

To investigate the possible intrinsic and observable kinematics of oblate elliptical galaxies, we construct self-consistent phase-space distribution functions (DFs), depending on three integrals of motion. Beginning with the two classical integrals E and Lz, and an approximative third integral obtained by Hamiltonian perturbation theory, we devise normalized orbit shape invariants Si. Rewriting the DF in terms of energy and two such alternative integrals makes it easy to visualize the physical orbit distribution that a particular DF stands for. We build the desired anisotropy into a galaxy model by writing the DF as a product of an assigned function h(Si) and a derived function g(E, Sj), selecting the assigned part of the DF, and solving for the derived part. We thus construct a variety of self-consistent models for a particular flattened isochrone potential; two-integral (2I) or three-integral, radially or tangentially anisotropic, rotating or not. We find that there is large freedom not only in the DF, but also in the resulting kinematic structure it implies. This we investigate by calculating rotation and dispersion curves as well as line-of-sight velocity profiles (VPs). Some more specific results are as follows. (i) Oblate galaxies may be flattened by either high |$\overline{\upsilon^{2}_{\phi}}$|⁠, or low |$\sigma_{\theta}$|⁠, or gradients in radial anisotropy. The second case occurs in a new 2I model in which |$\sigma_{r}\approx\sigma_{\phi}\gt\sigma_{\theta}$|⁠. (ii) The velocity anisotropy on the intrinsic minor axis is not strongly coupled to that on the major axis. There exist models radially anisotropic on the major axis which have isotropic, radial or tangential velocity ellipsoids on the minor axis. (iii) The velocity dispersion on the projected minor axis can thus be higher or lower than that on the major axis; this freedom reflects the third integral. (iv) Some rapidly rotating models show central gradients in the projected rotation and dispersion velocities reminiscent of those seen in some galactic nuclei. (v) The outer major axis VPs in our non-rotating 2I model are clearly double-peaked; those in the isotropic rotator are asymmetric about their velocity peaks. (vi) The recognition of the DF in terms of the VPs it produces on both axes is generally a subtle matter and requires further quantitative analysis. Results from spherical systems, however, suggest that such analysis will provide new information about the anisotropy and mass distribution in elliptical galaxies, and - because of the large number of self-consistent DFs available even for oblate systems - probably also about the processes which formed ellipticals