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

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