On the kinematic deconvolution of the local neighbourhood luminosity function

A method for inverting the statistical star counts equation, including proper motions, is presented; in order to break the degeneracy in that equation it uses the supplementary constraints required by dynamical consistency. The inversion gives access to both the kinematics and the luminosity function of each population in three régimes: the singular ellipsoid, the constant ratio Schwarzschild ellipsoid plane parallel models and the epicyclic model. This more realistic model is taylored to account for local neighbourhood density and velocity distribution. The first model is fully investigated both analytically and via means of a non-parametric inversion technique, while the second model is shown to be formally its equivalent. The effect of noise and incompleteness in apparent magnitude is investigated. The third model is investigated via a 5D+2D non-parametric inversion technique where positivity of the underlying luminosity function is explicitely accounted for. It is argued that its future application to data such as the Tycho catalogue (and in the upcoming satellite GAIA) could lead – provided the vertical potential, and/or the asymmetric drift or w are known – to a non-parametric determination of the local neighbourhood luminosity function without any reference to stellar evolution tracks. It should also yield the proportion of stars for each kinematic component and a kinematic diagnostic to split the thin disk from the thick disk or the halo.