4. DYNAMICAL ANALYSIS OF KINEMATIC DATA

This Section gives a brief overview of how the gravitational potential of an elliptical galaxy can be inferred from stellar-kinematic data. These data may include PN velocities, which individually sample the stellar line-of-sight velocity distributions L(v) (hereafter LOSVD) at their positions in the galaxy image, as well as ALS spectroscopy measurements of the first few moments of the LOSVD, at a set of 1D or 2D binned positions.

As is well known, velocity dispersion profile measurements (and streaming velocities, if the galaxy rotates) do not suffice to determine the distribution of mass with radius, due to the degeneracy with orbital anisotropy. With very extended measurements a constant M / L model can be ruled out (e.g., [18]), but the detailed M(r) still remains undetermined. Thus mass determination in elliptical galaxies always involves determining the orbital structure at the same time, and requires a lot of data. LOSVDs from absorption line profile shapes provide additional data with which the degeneracy between orbit structure and mass can largely be broken. Simple spherical models are useful to illustrate this [24]: at large radii, radial orbits are seen side-on, resulting in a peaked LOSVD (positive Gauss-Hermite parameter h₄), while tangential orbits lead to a flat-topped or double-humped LOSVD (h₄ < 0). Similar considerations can be made for edge-on or face-on disks [25] and spheroidal systems [26].

One may think of the LOSVDs constraining the anisotropy, after which the Jeans equations can be used to determine the mass distribution. However, the gravitational potential influences not only the widths, but also the shapes of the LOSVDs (see illustrations in [24]). Furthermore, eccentric orbits visit a range of galactic radii and may therefore broaden a LOSVD near their pericentres as well as leading to outer peaked profiles. Thus, in practice, the dynamical modelling to determine the orbital anisotropy and gravitational potential must be done globally, and is typically done in the following steps:

(0) choose geometry (spherical, axisymmetric, triaxial);

(1) choose dark halo model parameters, and set total luminous plus dark matter potential ;

(2) write down a composite distribution function (DF) f = _k a_k f_k, where the f_k can be orbits, or DF components such as f_k(E, L²), with free a_k;

(3) project the f_k to observed space, p_jk = K_j f_k d, where K_j is the projection operator for observable P_j, and denotes the line-of-sight coordinate and the velocities;

(4) fit the data P_j = _k a_k p_jk for all observables P_j simultaneously, minimizing a ² or negative likelihood, and including regularization to avoid spurious large fluctuations in the solution. This determines the a_k, i.e, the best DF f, given , which must be f > 0 everywhere;

(5) vary , go back to (1), and determine confidence limits on the parameters of .

If the mass distribution and gravitational potential are known from analysis of, e.g., X-ray data, step (5) can be omitted. Because this eliminates the degeneracy with orbital anisotropy, considerably fewer data are then needed.

Such a scheme has been employed regularly to model spherical and axisymmetric galaxies [e.g., 4, 27], using mostly orbits ("Schwarzschild's method") and, more rarely, distribution function components. Discrete velocity data were modelled in [e.g., 5, 17]. The modelling techniques used to constrain black hole masses from nuclear kinematics and dark halo parameters from extended kinematics are very similar.

Line-profile shape parameter measurements are now available for many nearby ellipticals, but those reaching to ~ 2R_e are still scarce [e.g., 28, 4]. Modelling of the outer mass profiles of ellipticals from such data has been done for some two dozen round galaxies in the spherical approximation, and for a few cases using axisymmetric three-integral models.

From recent SAURON integral field measurements it has become clear that the kinematics of many elliptical galaxies show features imcompatible with axisymmetry [16]. This has motivated Verolme et al. to extend Schwarzschild's method to triaxial models [29], a development that has only recently become possible with the increase of available computing power. In parallel, adaptive N-body codes are being developed that use the algorithm of Syer & Tremaine for training N-body systems to adapt to a specified set of data constraints [30]. These latter models have the additional advantage of simultaneously providing a check for the model's dynamical stability. One application has been to the rotating Galactic bar; work currently in progress is on axisymmetric and rotating triaxial galaxies.