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 profiles and (spherical) total mass profiles, without the need for any explicit deprojection. We then show that an almost flat rotation curve, combined with modest velocity anisotropy, is already sufficient to recover the kinematic profiles 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 specific location 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 $R_{\rm e}$ can be estimated as $2.4 R_{\rm e} \langle \sigma^{2}_{\rm p} \rangle/ G$, where $\langle \sigma^2_{\rm p} \rangle$ 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 apertures 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 (within $\approx$ 10% accuracy). 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.