Charting Galactic Accelerations: When and How to Extract a Unique Potential from the Distribution Function

TL;DR

This paper presents a novel method to uniquely determine galactic accelerations from the distribution function of stars using the collisionless Boltzmann equation, validated with mock datasets and applicable to real stellar kinematic data.

Contribution

It introduces a semi-analytic approach leveraging normalizing flows and the Boltzmann equation to extract gravitational accelerations from six-dimensional stellar data.

Findings

Accurate acceleration extraction from mock datasets with noise

Method works for isotropic and anisotropic models

Provides a robust framework for analyzing stellar kinematics

Abstract

The advent of datasets of stars in the Milky Way with six-dimensional 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 datasets of discrete positions and velocities of stars are then discussed. Following Green & Ting (arXiv:2011.04673), 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 over-constrained linear equations provided by the collisionless Boltzmann equation. Testing our algorithm on mock datasets 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 six-dimensional stellar kinematics of an equilibrium system.