Dynamical models of elliptical galaxies - I. Simple methods

We study dynamical models for elliptical galaxies, deriving the projected kinematic profiles in a form that is valid for general surface brightness laws and (spherical) total mass profiles, without the need for any explicit deprojection. We provide accurate approximations of the line of sight and aperture-averaged velocity dispersion profiles for galaxies with total mass density profiles with slope near -2 and with modest velocity anisotropy using only single or double integrals, respectively. This is already sufficient to recover many of the kinematic properties of nearby ellipticals. As an application, we provide two different sets of mass estimators for elliptical galaxies, based on either the velocity dispersion at a location at or near the effective radius, or the aperture-averaged velocity dispersion. In the large aperture (virial) limit, mass estimators are naturally independent of anisotropy. The spherical mass enclosed within the effective radius R-e can be estimated as 2.4 R-e{/G, where is the average of the squared velocity dispersion over a finite aperture. This formula does not depend on assumptions such as mass-follows-light, and is a compromise between the cases of small and large aperture sizes. Its general agreement with results from other methods in the literature makes it a reliable means to infer masses in the absence of detailed kinematic information. If on the other hand the velocity dispersion profile is available, tight mass estimates can be found that are independent of the mass-model and anisotropy profile. In particular, for a de Vaucouleurs surface brightness, the velocity dispersion measured at approximate to 1R(e) yields a tight mass estimate (with 10 per cent accuracy) at approximate to 3R(e) that is independent of the mass model and the anisotropy profile. This allows us to probe the importance of dark matter at radii where it dominates the mass budget of galaxies. Explicit formulae are given for small anisotropy, large radii and/or power-law total densities. Motivated by recent observational claims, we also discuss the issue of weak homology of elliptical galaxies, emphasizing the interplay between morphology and orbital structure.