The Photo-Astrometric Vertical Tracer Density of the Milky Way I: The Method

We introduce a method to infer the vertical distribution of stars in the Milky Way using a Poisson likelihood function, with a view to applying our method to the Gaia catalogue. We show how to account for the sample selection function and for parallax measurement uncertainties. Our method is validated against a simulated sample drawn from a model with two exponential discs and a power-law halo profile. A mock Gaia sample is generated using the Gaia astrometry selection function, whilst realistic parallax uncertainties are drawn from the Gaia Astrometric Spread Function. The model is fit to the mock in order to rediscover the input parameters used to generate the sample. We recover posterior distributions which accurately fit the input parameters to within statistical uncertainties, demonstrating the efficacy of our method. Using the GUMS synthetic Milky Way catalogue we find that our halo parameter fits can be heavily biased by our overly simplistic model, however, the fits to the thin and thick discs are not significantly impacted. We apply this method to Gaia Early Data Release 3 in a companion paper where we also quantify the systematic uncertainties introduced by oversimplifications in our model.