Dynamical and gravitational lensing properties of a new phenomenological model of elliptical galaxies

Aims. Recent observations of the line of sight velocity profile of elliptical galaxies have furnished controversial results with some works favouring the presence of a large amount of dark matter in the outer regions and others arguing in favour of no dark matter at all. In order to shed new light on this controversy, we propose here a new phenomenological description of the total mass profile of galaxies. Methods. Under the hypothesis of spherical symmetry, we assume a double power-law expression for the global mass-to-light (hereafter $M/L$) ratio $\Upsilon(r) = M(r)/L(r) = \Upsilon_0 (r/r_0)^{\alpha} (1 + r/r_0)^{\beta}$ with $(\Upsilon_0, r_0)$ scaling parameters and $(\alpha, \beta)$ determining the inner and outer slope of the $M/L$ ratio. In particular, $\Upsilon \propto r^{\alpha}$ for $r/r_0 \ll 1$ so that $\alpha = 0$ mimics a constant $M/L$ ratio in the inner regions, while, for $(r/r_0 \gg 1)$, $\Upsilon \propto r^{\alpha + \beta}$ thus showing that models with $\alpha + \beta = 0$ have an asymptotically constant $M/L$ ratio. A wide range of possibilities is obtained by varying the slope parameters $(\alpha, \beta)$ in the range we determine on the basis of physical considerations. Choosing a general expression for the luminosity density profile $j(r)$, we work out an effective galaxy model that accounts for all the phenomenology observed in real elliptical galaxies. We derive the main dynamics and gravitational lensing properties of such an effective model. Results. We analyze a general class of models, able to take into account different dynamical trends. We are able to obtain analytical expressions for the main dynamical and lensing quantities. We show that constraining the values of $\alpha + \beta$ makes it possible to analyze the problem of the dark matter in elliptical galaxies. Indeed, positive values of $\alpha + \beta$ would be a strong evidence for dark matter. Conclusions. Finally we indicate possible future approaches in order to face the observational data, in particular using velocity dispersion profiles and lensed quasar events.