A method to calculate gravitational accelerations within discrete localised regions in the Milky Way

We present a method to calculate gravitational potential gradients within regions containing few tens of thousands stars with known phase space coordinates. The central idea of the method is to calculate orbital arcs for each star within a given region for a certain parametrised potential (gravitational acceleration) and to assume that position of each star on its orbital arc is a random variable with a uniform probability density in time. Thereafter, by combining individual probability densities of stars it is possible to calculate the overall probability density distribution and likelihood for a given region as a function of gravitational acceleration parameters. The likelihood has a maximum if the calculated probability distribution and the observed distribution of stars in phase space are consistent. This allows us to constrain gravitational accelerations and potential gradient values. The method assumes that phases of stars are mixed within the regions where stellar orbits are calculated. We tested the method for 12 small rectangular regions within simulated disc galaxy from Gaia Wiki. Tests show that even with a rather simple acceleration form the calculated accelerations in galactic plane coincide with their true values from simulation about 5 per cent, misalignment between the calculated and true acceleration vector directions is less than 1 degree (median values). The model can be used with the Milky Way Gaia complete solution data.