Non-parametric reconstruction of distribution functions from observed galactic discs

A general inversion technique for the recovery of the underlying distributi on function for observed galactic discs is presented and illustrated. Under the assumption that these discs are axisymmetric and thin, the proposed me thod yields a unique distribution compatible with all the observables avail able. The derivation may be carried out from the measurement of the azimuth al velocity distribution arising from positioning the slit of a spectrograp h along the major axis of the galaxy. More generally, it may account for th e simultaneous measurements of velocity distributions corresponding to slit s presenting arbitrary orientations with respect to the major axis. The app roach is non-parametric, i.e. it does not rely on a particular algebraic mo del for the distribution function. Special care is taken to account for the fraction of counter-rotating stars, which strongly affects the stability o f the disc. \ An optimization algorithm is devised - generalizing the work of Skilling & Bryan - to carry this truly two-dimensional ill-conditioned i nversion efficiently. The performance of the overall inversion technique wi th respect to the noise level and truncation in the data set is investigate d with simulated data. Reliable results are obtained up to a mean signal-to -noise ratio of 5, and when measurements are available up to 4(Re). A discu ssion of the residual biases involved in non-parametric inversions is prese nted. The prospects of application of the algorithm to observed galaxies an d other inversion problems are discussed.