Three-integral oblate galaxy models

Abstract

A simple numerical scheme is presented for the construction of three-integral phasespace distribution functions for oblate galaxy models with a gravitational potential of St ackel form, and an arbitrary axisymmetric luminous density distribution. The intrinsic velocity moments can be obtained simultaneously with little extra e ort. The distribution of the inner and outer turning points of the short-axis tube orbits that are populated can be speci ed freely, and is chosen in advance. The entire distribution function is then derived from the density by an iterative scheme that starts from the explicitly known distribution function of the thin-orbit (maximum streaming) model, in which only the tubes with equal inner and outer turning points are populated. The versatility and limitations of this scheme are illustrated by the construction of a number of self-consistent three-integral attened isochrone models of Kuzmin{Kutuzov type, and by investigation of special cases where the scheme is tractable analytically. This includes the behaviour of the distribution functions in the outer regions of the models. The scheme converges rapidly for models containing orbits with ratios of the outer to inner turning point as large as ten, and is particularly suited for the construction of tangentially anisotropic attened models, self-consistent as well as non-consistent. The algorithm simpli es in the disk and spherical limit, and can be generalized to triaxial models.

Elliptical Galaxy Dynamics

“…Sellwood & Valluri found stability to axisymmetric modes at an axis ratio of ∼ 1 : 4 in their models. Ringlike modes in axisymmetric models were also reported by and Robijn (1995).…”

“…Sellwood & Valluri found that the lopsided modes became less important as the net rotation of their models was increased, but they persisted even in models with net angular momentum 90% that of a maximally rotating model. Robijn (1995) carried out a normal-mode analysis of Kuz'min-Kutuzov models with little radial pressure and confirmed the Merritt & Stiavelli result that such models were unstable to lopsided modes regardless of axis ratio. He found that the growth rate of lopsided modes was more strongly affected by increasing the radial velocity dispersion than by adding net rotation.…”

Spherical symmetry breaking in cold gravitational collapse of isolated systems

“…It was long believed that elliptical galaxy is oblate due to the flattening about the axis of rotation. Theoretical supports for oblate galaxy could be found in Dehnen & Gerhard (1994); Robijn & de Zeeuw (1996) and also references therein. An objection was then made based on the observed rotational velocity of galaxies that led to another propositions for prolate galaxy (e.g.…”

General solution of the Jeans equations for triaxial galaxies with separable potentials

“…Two more follow from the fact that T φφ is perpendicular to the (complete) z ‐axis, and thus coincides with T λλ and T νν on the part between and beyond the foci, respectively: For oblate models with thin S‐tube orbits ( T λλ ≡ 0, see ), the analytical solution of was derived by Bishop (1987) and by de Zeeuw & Hunter (1990). Robijn & de Zeeuw (1996) obtained the second‐order velocity moments for models in which the thin‐tube orbits were thickened iteratively. Dejonghe & de Zeeuw (1988, Appendix D) found a general solution by integrating along characteristics.…”

“…For oblate models with thin S-tube orbits (T λλ ≡ 0, see §2.5.6), the analytical solution of (2.22) was derived by Bishop (1987) and by de Zeeuw & . Robijn & de Zeeuw (1996) obtained the second-order velocity moments for models in which the thin tube orbits were thickened iteratively. Dejonghe & de Zeeuw (1988, Appendix D) found a general solution by integrating along characteristics.…”