Extended distribution functions for our Galaxy

We extend models of our Galaxy based on distribution functions (DFs) that are analytic functions of the action integrals to extended distribution functions (EDFs), which have an analytic dependence on metallicity as well. We use a simple, but physically-motivated, functional forms for the metallicity of the interstellar medium as a function of radius and time and for the star-formation rate, and a model for the diffusion of stars through phase space to suggest the required functional form of an EDF. We introduce a simple prescription for radial migration that preserves the overall profile of the disc while allowing individual stars to migrate throughout the disc. Our models explicitly consider the thin and thick discs as two distinct components separated in age. We show how an EDF can be used to incorporate realistic selection functions in models, and to construct mock catalogues of observed samples. We show that the selection function of the Geneva-Copenhagen Survey (GCS) biases in favour of young stars, which have atypically small random velocities. With the selection function taken into account our models produce good fits of the GCS data in chemo-dynamical space and the Gilmore and Reid (1983) density data. From our EDF, we predict the structure of the SEGUE G-dwarf sample. The kinematics are successfully predicted. The predicted metallicity distribution has too few stars with [Fe/H]$\simeq-0.5$ dex and too many metal-rich stars. A significant problem may be the lack of any chemical-kinematic correlations in our thick disc. We argue that EDFs will prove essential tools for the analysis of both observational data and sophisticated models of Galaxy formation and evolution.