Annu. Rev. Astron. Astrophys. 1991. 29: 239-274
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2.1 Spherical and Axisymmetric Models

A spherical potential admits four isolating integrals of motion: the orbital energy E and the three components of the angular momentum vector L = (Lx, Ly, Lz). One usually considers models without any preferred axis, so that f can depend only on |L|, i.e., f = f (E, L2). Such models have anisotropic velocity distributions. It is possible to find the unique f = f (E) that generates a given spherical density rho (r) by means of Eddington's inversion formula (103). If rho (r) falls off with radius sufficiently rapidly, this f (E) is nowhere negative, and represents the unique isotropic model. Anisotropic models can sometimes be found by similar inversion methods (70), but they are usually constructed by assumption of a special functional form for f (E, L2) (e.g., 139, 236, 247, 323) or by numerical techniques (280, 290, 294). Although few elliptical galaxies are spherical (43, 303) there is ample room for further work on the properties of anisotropic models. In particular, much information is contained in the line profiles, i.e., the distribution of stars as a function of line-of-sight radial velocity at different projected distances from the center. These profiles can now be measured (25, 122), but so far few theoretical profiles have been calculated (71).

Axisymmetric potentials admit two exact isolating integrals, the energy E and one angular momentum component, Lz say. Lynden-Bell (219) generalized Eddington's formula, and showed that for a given axisymmetric density rho (R, z) there is a unique distribution function f (E, Lz2). In such two-integral models the velocity dispersion < vR2 > in the equatorial plane is always equal to the perpendicular dispersion < vz2 >. The method for the actual calculation of f is not easy to apply in practice, and only a few models have been constructed in this manner (74, 161, 62, 201). Other approaches include assuming a functional form for f, and using either series expansions (199, 259), or a numerical technique (202, 75) to find the associated density. The latter has been used recently to model the velocity fields of bulges of spiral galaxies (177, 178, 296). More details on axisymmetric models can be found in references (70, 82, 84).

Orbits with Lz neq 0 have a definite sense of rotation around the symmetry axis, but both clockwise and counterclockwise motion may occur. It follows that non-zero mean streaming is possible around the symmetry axis of any axisymmetric model, and that its magnitude is not fixed by the density, since in each orbit the fraction of direct versus retrograde stars may be chosen freely. In two-integral models < vphi > is often chosen such that the azimuthal dispersion < vphi2 > - < vphi >2 = < vR2 > = < vz2 > (e.g., 302). Models of this kind are referred to as isotropic rotators. Other choices for < vphi > are discussed by Dejonghe (70).

Most orbits in realistic axisymmetric potentials are tubes around the symmetry axis, and possess an extra isolating integral of motion, I3 (e.g., 64, 226, 288). The remaining phase-space is generally made up of a host of minor orbit families and irregular orbits. The latter do not have a third integral, and Binney has argued that for this reason Jeans' theorem is not valid for such systems (40, 275), implying that no true equilibrium solutions may exist. In many cases of interest the fraction of irregular orbits turns out to be small, and they are nearly indistinguishable from regular orbits on time scales of the order of a Hubble time. For practical purposes one may therefore probably still use Jeans' theorem, and construct approximate equilibrium models. The potentials relevant for elliptical galaxies are nearly spherical, and I3 is related closely to L2 (172, 298). This fact has been used to construct approximate three-integral models with density profiles appropriate for elliptical galaxies (271, 272). Models based on special potentials which support an exact third integral are discussed in Section 2.3.

Kinematic properties of dynamical models are often calculated by direct solution of the Jeans equations which relate the velocity dispersions to the density and the potential of the model (48, 180). Various solutions for spherical systems have been used to model the kinematics of elliptical galaxies, e.g., in connection with the search for nuclear black holes (45, 329, Section 3.5.1). An explicit solution of the Jeans equations is available also for two-integral axisymmetric models (163), but anisotropic axisymmetric solutions cannot be derived unless certain assumptions are made (7, 9, 116, Section 3.3.2). The resulting dispersion profiles are not always guaranteed to correspond to a distribution function f geq 0, however

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