Abstract
ABSTRACT 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.
Highlights
In galactic astronomy, a fundamental problem is to extract the underlying gravitational potential from the kinematics of a tracer population
The phase-space distribution (DF) for the stars in the Milky Way is an obvious way to organize the new datasets comprising of nearby stars in the full six-dimensional phase-space coordinates
One question that follows is what information the distribution function (DF) contains about the overall properties of the Galaxy
Summary
A fundamental problem is to extract the underlying gravitational potential from the kinematics of a tracer population. Green & Ting (2020) recently raised the possibility of direct determination of the gravitational potential from the distribution function using the collisionless Boltzmann equation itself. This is the continuity equation satisfied by the distribution function in the six-dimensional phase space of positions and velocities. At every location in physical space, the collisionless Boltzmann equation provides a single constraint on the three unknown components of the gravitational force It is unclear if the identification of a stationary distribution function is sufficient to specify uniquely the gravitational potential (modulo an additive constant). We demonstrate the efficacy of our method on mock datasets sampled from isotropic and anisotropic distribution functions of galaxy models, including the effects of errors
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