Line-of-sight velocity profiles in spherical galaxies: breaking the degeneracy between anisotropy and mass

Abstract

A systematic study of the line profiles expected in the outer parts $$(R \gg {R}_{c})$$ of non-rotating elliptical galaxies is reported. For a family of quasi-separable spherical models the line-of-sight velocity profiles $$l(v)$$ are calculated analytically in both a Keplerian and a scale-free halo potential. These velocity profiles (VPs) can take a wide variety of shapes; they illustrate that tangentially anisotropic distribution functions (DFs) generally lead to VPs with flat tops, while radially anisotropic DFs lead to more nearly Gaussian VPs. Strongly radial DFs in a halo potential develop prominent wings. As a general rule these velocity profiles are sensitive to the anisotropy β of the model, less so to the stellar density profile and the potential, and least so to the form of the DF at fixed β. The velocity dispersion σ is sensitive to anisotropy, stellar density, and potential. Thus the best strategy to infer the mass distribution in elliptical galaxies from kinematic data is to estimate β from the VPs and then use the dispersion profile to constrain the potential. In any case, in a Keplerian potential the velocity dispersion can exceed the isotropic value by only ~ 10–15 per cent for reasonably tangential DFs. These VPs are then quantitatively analysed by means of a new set of Gauss-Hermite moments, which arise from an expansion in terms of corresponding orthogonal functions. These moments are much less sensitive to uncertain power in the profile wings than are the classical moments, and with increasing order they describe structure of increasing frequency in the VP. The zeroth-order moment s₀ represents a Gaussian with dispersion σ; odd and even moments s_n, describe asymmetric and symmetric deviations from a Gaussian profile shape, respectively. Analysis of the quasi-separable VPs in terms of the new moments shows a tight relation between the second moment s₂ and the anisotropy parameter $$\beta\enspace \text {for}\enspace R \gg {R}_{c}$$, which depends only weakly on the potential and stellar density slope. For tangentially anisotropic DFs s₂+ s₄ appears to be a good indicator of the potential. I present a set of diagrams that can be used to estimate β and the circular velocity $$v_c$$ from measured Gauss-Hermite moments s₂ and the dispersion profile. If a halo is thus detected, and the line profiles are near- Gaussian down to 20 per cent of peak amplitude, this limits the degree of radial anisotropy. By contrast, even large changes in the DF lead to only slight variations in the VPs as long as the model's anisotropy β is kept constant. One example is a family of spherical models with $$\rho \propto {r}^{-2}$$ and exactly β = 0 which includes the classical singular isothermal sphere.