Charting galactic accelerations: when and how to extract a unique potential from the distribution function

The advent of data sets of stars in the Milky Way with 6D phase-space information makes it possible to construct empirically the distribution function (DF). Here, we show that the accelerations can be uniquely determined from the DF using the collisionless Boltzmann equation, providing the Hessian determinant of the DF with respect to the velocities is non-vanishing. We illustrate this procedure and requirement with some analytic examples. Methods to extract the potential from data sets of discrete positions and velocities of stars are then discussed. Following Green & Ting, we advocate the use of normalizing flows on a sample of observed phase-space positions to obtain a differentiable approximation of the DF. To then derive gravitational accelerations, we outline a semi-analytic method involving direct solutions of the overconstrained linear equations provided by the collisionless Boltzmann equation. Testing our algorithm on mock data sets derived from isotropic and anisotropic Hernquist models, we obtain excellent accuracies even with added noise. Our method represents a new, flexible, and robust means of extracting the underlying gravitational accelerations from snapshots of 6D stellar kinematics of an equilibrium system.